The Speyer School

The Speyer School (1935-1941), was started within a building at Columbia's Teachers College, and named after financier James Speyer in 1902. The Speyer School became famous between 1935-39 when it was used a "laboratory" to study how children perform when separated by educational ability. There were seven classrooms with 175 students, which had an average IQ on the Stanford Binet test of between 75-90 and two classrooms with students that tested at the level of 130+ on the same IQ test.

The experiment was led by Leta Stetter Hollingworth, an American psychologist who specialized in education and who is credited with creating the term "gifted children." The results of the several studies conducted by Hollingworth dispelled many myths about exceptional children and suggested that many exceptionally gifted children suffer adjustment problems due to two factors: inept treatment by adults and lack of intellectual challenge. Her work also made “emotional education” an important part of the original Speyer School curriculum.

References

Category:Columbia University


DENVER (AP) — An erratic wildfire charging through extremely dry land in the heart of Colorado ski country destroyed three homes and forced people to flee, authorities said Thursday.

Elsewhere in Colorado, a rare high-elevation tornado touched down at the site of another wildfire Thursday but apparently caused little damage and had no effect on the fire.

They were among more than 60 large blazes burning across the United States, mostly in the West, where whipping winds and increasing heat have made it easy for flames to spread.

Fires exploded in Northern California, Utah and other areas, where a prolonged and severe drought has desiccated forests.

In Colorado, residents of multimillion-dollar properties, modest condos and mobile homes were ordered to evacuate early Thursday because of an unpredictable wildfire reported the night before near the town of Basalt. More than 500 homes were affected.

The Eagle County Sheriff’s Department said no other homes had been lost besides the three reported earlier.

Smoke from the fire temporarily halted flights into the resort town of Aspen, about 20 miles (32 kilometers) away.

Authorities believe the flames started after people at a shooting range used tracer bullets, which illuminate the path of fired bullets.

A tornado was reported Thursday south of Fairplay, a central Colorado town about 10,000 feet (3,000 meters) above sea level. It appeared to touch down at or near the edge of a wildfire that has burned about 17 square miles (43 square kilometers), National Weather Service meteorologist Russell Danielson.

Tornadoes are rare at that elevation and are seldom seen at any wildfire, Daniels said.

Another Colorado wildfire about 205 miles (330 kilometers) southwest of Denver destroyed more than 130 homes and forced more than 2,000 people in three counties to evacuate. It grew to about 160 square miles (nearly 420 square kilometers) Thursday.

Rain was expected in parts of Colorado, which could dampen fires but trigger flash flooding in areas left bare by wildfires.

Hotter, drier weather was forecast in Northern California, where thousands of firefighters were battling a wildfire that was already about three times the size of San Francisco.

By Thursday night, the fire northwest of Sacramento had burned nine buildings and 137.5 square miles (356 square kilometers). However, authorities didn’t immediately indicate whether any homes had been destroyed in the rural area, although about 110 structures remained threatened, according to state fire officials.

The blaze was 33 percent contained.

“The cool weather on Wednesday helped firefighters, but it will become hotter and drier starting Friday and into the weekend, and that will be a challenge,” said Anthony Brown, a spokesman with the California Department of Forestry and Fire Protection.

Some of the 2,500 people who were under mandatory evacuations orders were allowed to return home.

Authorities say a fire that jumped across Interstate 5 south of the Oregon border in California has closed the southbound lanes in that area.

The Oregon Department of Transportation said Thursday evening the southbound lanes would remain closed at Ashland, Oregon. Motorists were advised to take an alternate route toward California.

California Gov. Jerry Brown declared a state of emergency for Siskyou County, citing “extreme peril” to people and property. There were some evacuation in the tiny town of Hornbrook.

The fire called Klamathon started Thursday afternoon.

The California Department of Forestry and Fire Protection said Thursday night that the blaze had burned nearly 8 square miles (20 square kilometers). Multiple structures are threatened, damaged or destroyed, the department said.

In northern New Mexico, a wildfire closed a ranch where novelist D.H. Lawrence once sought spiritual renewal. Officials said a fire in drought-stricken Carson National Forest has scorched nearly 4 square miles (10 square kilometers) since June 24 and is only partially contained.

Forest restrictions imposed last week closed the University of New Mexico’s D.H. Lawrence Ranch, which hosted the writer in the summers of 1924 and 1925 as well as Swiss psychiatrist Carl Jung, author Willa Cather and artist Georgia O’Keeffe after Lawrence’s 1930 death.

In Utah, scorching summer temperatures and winds quickly pushed flames through bone-dry vegetation near a popular fishing lake about two hours southeast of Salt Lake City. The 66-square-mile (171-square-kilometer) fire near Strawberry Reservoir has burned about 90 structures.

Darren Lewis’ family cabin is in the fire’s path and he fears the A-frame structure built 46 years ago by his uncle and father has burned, though he’s waiting for confirmation.

Lewis, 44, and his extended family planned to spend the Fourth of July at the property in a narrow canyon near a river but instead nervously watched social media for updates from the Salt Lake City suburb of Magna.

More than 1,100 people have been evacuated in the mountain area, and the family hasn’t been able to go back since Monday, he said. It is a frequent family gathering spot and has special sentimental value because his father and uncle both died last year.

“That’s our healing place,” Lewis said.

___

Associated Press writers Russell Contreras in Albuquerque, Dan Elliott in Denver, Brady McCombs in Salt Lake City and Olga R. Rodriguez in San Francisco contributed to this report.


Q:

Как в html поместить иконку SVG?

Не отображает иконку SVG. Если задать bacgkround-image, то иконка отображается (т.е. путь указан правильно). Как правильно прописать ее в HTML? Или кто подскажет другой способ? Иконку взяла с material design icons, в  Angular Material подходящей нет. Доступ по url иконки не подходит, она должна храниться локально

    <button>

    </button>

Код иконки

<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" focusable="false" width="2em" height="2em" style="-ms-transform: rotate(360deg); -webkit-transform: rotate(360deg); transform: rotate(360deg);" preserveAspectRatio="xMidYMid meet" viewBox="0 0 24 24"><path d="M12 5H9l4-4 4 4h-3v4.43a8.638 8.638 0 0 0-2 1.69V5zm-1.6 10H8.5l-.7-2H4.6l-.7 2H2l3.2-9h2l3.2 9zm-3.05-3.35L6.2 8l-1.15 3.65h2.3zM23 11l-4-4v3a6.75 6.75 0 0 0-7 6.17 3 3 0 1 0 2 0c.47-4 4.47-4.2 4.95-4.2v3L23 11z" /></svg>

A:

Добавил скаченную иконку на сервер и загрузил её в HTML с помощью

<button>

 </button>

Значит у вас проблемы с путём до файла *.svg
Но в случае добавления  иконки с помощью  она ведёт себя как обыкновенное изображение, поэтому затруднительно сделать стилизацию обычными приемами с помощью CSS.
Пока нет достаточного опыта работы с SVG, используйте инлайновый способ добавления иконки.
Удалил из кода иконки fill="#626262" чтобы не мешало стилизации иконки

.btn {
fill:dodgerblue;
transition:fill 0.5s ease;
}
.btn:hover {
fill:crimson;
}
<button class="btn">
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" focusable="false" width="2em" height="2em" style="-ms-transform: rotate(360deg); -webkit-transform: rotate(360deg); transform: rotate(360deg);" preserveAspectRatio="xMidYMid meet" viewBox="0 0 24 24"><path d="M12 5H9l4-4 4 4h-3v4.43a8.638 8.638 0 0 0-2 1.69V5zm-1.6 10H8.5l-.7-2H4.6l-.7 2H2l3.2-9h2l3.2 9zm-3.05-3.35L6.2 8l-1.15 3.65h2.3zM23 11l-4-4v3a6.75 6.75 0 0 0-7 6.17 3 3 0 1 0 2 0c.47-4 4.47-4.2 4.95-4.2v3L23 11z" /></svg>
</button>

Ещё способ добавить svg иконки
Данная техника подойдет, если нужно быстро добавить иконки в свой проект, которые можно раскрасить в любой цвет из внешнего файла CSS.
  Причём иконки высокого качества, практически полный набор на все случаи.
 Иконки адаптивны и не теряют качества при масштабировании, так как это иконки SVG из которых состоит символьный шрифт  - Google Material Icons
Подключение шрифта:
<link href="https://fonts.googleapis.com/icon?family=Material+Icons"
      rel="stylesheet">

Выбирать иконки по названиям можно здесь
Выбираем нужную иконку с именем - favorite
Подключение иконки к HTML странице:
favorite, где
material-icons - общий класс для всех иконок
red - дополнительный класс, задающий цвет окраски иконки
Ниже пример с тремя одинаковыми иконками, которые раскрашены в разные цвета
Добавлена анимация увеличения иконок при hover.

.material-icons {
transform: scale(2);
}
i {
  margin: 100px 50px 50px 80px;
  transition: transform 0.5s ease-in-out;
}
i:hover {
  transform: scale(4);
}

.red{color:red}
.green{color:yellowgreen}
.blue{color: dodgerblue}
<link href="https://fonts.googleapis.com/icon?family=Material+Icons"
      rel="stylesheet">
favorite
android
backup

Размер иконок также можно менять, здесь удвоенный исходный размер:
.material-icons {
transform: scale(2);
}


Ave built a six-second lead in the opening seven laps over Lopez, Drissi and
Pickett. On lap eight, Lopez began to separate his No. 06
WabashPower.com/Mobil 1/Goodyear Chevrolet Corvette from the pair of Jaguars
and started to reel in Ave's Mustang. Ten laps later, Lopez was all over Ave
and took the lead on lap 20.

From there, Ave fell back into the clutches of the pair of Jaguars,
eventually dropping from second to fourth on lap 24, with Pickett moving to
second and Drissi remaining third.

All the while, Rookie Skeen, known to the road racing community from the
2008 television show Setup, was slicing through the field after starting
last (19th) in his No. 70 Atlantic Autosport Chevrolet Corvette. Two laps
later, he was clear of Ave with his sights set on the Jaguars of Drissi and
Pickett.

In the meantime, Lopez drove away to a four second lead and looked to be in
good shape to take a Trans-Am Series debut win. In traffic, Pickett was able
to bring the gap down to the two second range, but on lap 32, it was back up
to nearly three seconds. Eight corners later, it was all over, as the
Corvette's motor expired on the back straightaway without warning.

"It just went," Lopez said. "I was 100 percent sure I had the win. I was
starting to cruise, saw Pickett coming after the traffic and was able to
pick the pace back up. Next time..."

At the same time Lopez's motor was expiring, Ave tangled with another car
and ended up in the Turn One gravel, ending a run that started with his
first series pole earlier in the day. Both incidents brought out the
full-course caution and when the field came around to be picked up by the
pace car, Pickett was in front, with Drissi second and Skeen third.

As the cleanup ensued, it appeared that the final three would be locked in
with the 36-lap, 91.44-mile contest ending under yellow, but on the final
tour, Drissi's Jaguar sputtered to a stop with a fuel pickup problem,
promoting Skeen to second and Ruman's No. 30 McNichols/Goodyear/Cenweld Corp
Chevrolet Corvette to third.

"This is a little fairy tale, winning the first race in the series that
Muscle Milk sponsors," Pickett said. "I'm just tickled for the SCCA and our
Muscle Milk brand. Trans-Am is back!

"It was a difficult race. I've done a lot of these races, and you can't push
too hard -- it just won't work over the course of a race. RJ [Lopez] was going
faster than I wanted to go at that point. But he was gradually coming back
to me. It was too bad for him, but look -- we have two GT-1 cars on the podium.
That's fantastic. We need more people to bring their GT-1 cars out to race
with us in Trans-Am!"

Skeen earned the Flowmaster Star of the Race Award and the Sunoco Hard
Charger of the race for his drive from 19th to second. He set a new Trans-Am
Series lap record during his run to the front of 1:22.650 (110.635 mph).
Skeen was visibly exhausted after the race.

"This is our first race this long and I don't have a cool suit," Skeen said.
"In fact, we've only had the car a month and weren't really sure we were
coming until last week. In qualifying, we had a wheel separate and didn't
turn a time. We knew we would have a good pace because we were good in
practice and that was on old tires -- we only had one set of sticker tires.

"From the start, we just went for all the marbles. I knew I was catching the
Jaguars, and figured Tony [Ave] was running away. Toward the end, I was
starting to get hot, but I was still keeping up. It's great to finish
second."

Ruman started sixth and ran seventh for the first 20 laps before gradually
moving up as other cars retired. When, Drissi stopped on course on the final
lap, she wasn't sure what to do and didn't realize where she'd finished
until after the checkered flag fell.

"This is just awesome," Ruman, who equaled her father Bob Ruman's best
finish of third, said. "This McNichols team has worked so hard for so long.
[Jeff] Emery and I were battling for a long time and when I got clear, I was
about 14-15 seconds off the cars ahead of me and we were trying to close in
and were beginning to. It really fell our way today in the end."

Ruman's accomplishment was particularly impressive as she was competing with
a 310ci engine in a sea of larger-displacement powerplants.

John Schaller, of Fort Myers, Fla., moved from eighth on the grid to finish
fourth in his No. 08 JRS Motorsport/Central Power Distribution Chevrolet
Camaro, followed by Drissi, who was the first car one lap down.


Optimized culture conditions for bacteriocin production by Pediococcus acidilactici LAB 5 and its characterization.
A strain of Pediococcus acidilactici LAB 5 was isolated from vacuum-packed fermented meat product, in order to obtain a novel bacteriocin from food-grade organisms. Optimized culture conditions for bacteriocin production in different media (viz., MRS, TGE, TGE + buffer, TGE + Tween 80, and TGE + Tween 80 + buffer) and at different temperatures and pH conditions were reported. TGE + Tween 80 + buffer medium was found to be most effective for bacteriocin production (about 2400 AU/ml) by this strain, when incubated at 37 degrees C for 24 h. Bacteriocin, partially purified by adsorption-desorption method showed molecular mass of 10.3 kDa and produced prominent inhibition zone in activity gel. It showed significant storage stability both at high as well as in low temperatures for up to 6 months and retained its activity in a number of organic solvents, except in 2-mercaptoethanol. The treatment with amylase and lysozyme did not change its activity, but it lost its activity on proteinase K treatment. Antibacterial efficacy of bacteriocin was proved against some food spoilage and human pathogenic bacteria like Enterococcus, Leuconostoc, Listeria, Staphylococcus and Streptococcus.


1. Field of the Invention
The present invention relates to a technique for manufacturing liquid crystal display devices with a slim bezel by surface close attachment.
2. Description of the Background Art
In most cases, when manufacturing liquid crystal display devices, a plurality of panels are provided on a single mother glass, and then divided into individual panels. In recent years, in order to eliminate generation of small scraps to simplify the manufacturing steps, and to make efficient use of the mother glass, it has become increasingly common to manufacture liquid crystal display devices by closely attaching adjacent panels (hereinafter referred to as “surface close attachment”). Further, a bezel region constituting a non-display region around a panel tends to be made slim, and so-called bezel slimming has been promoted.
In this manner, when panels are formed by surface close attachment and bezel slimming becomes advanced, a region for forming a transfer member and a sealing member provided on a bezel region (for the sealing member, especially protrusions in which an inlet is provided) as well as a margin of distance for a cutting line for separating between panels and for adjacent panels are reduced.
In addition, such a transfer member and a sealing member are both typically formed in such a manner that a paste material that has been applied is sandwiched between substrates and then spread therebetween, and therefore positions for formation and amounts of spread vary to a relatively large extent.
Therefore, it has become difficult to form a transfer member and a sealing member in a predetermined region of formation for these members without making their paste materials spread out of the region of formation. If the paste materials spread out over the cutting line for separating between panels or over an adjacent panel region, the glass substrate may be adhered (bonded) by the paste material at a portion not desired, resulting in wrong cutting.
As described above, in the case in which panels are formed by surface close attachment and with a slimmer bezel, probability of a wrong cutting due to spreading of the paste materials increases, and consequently it is difficult to manufacture liquid crystal display devices at a high yield ratio.
Thus, regarding the spreading out of paste materials, Japanese Patent Application Laid-Open No. 2003-215622, International Publication No. WO00/45360, and Japanese Patent Application Laid-Open No. 2006-268020, for example, disclose a method of forming a transfer member a within a predetermined region by enclosing a region of formation for a transfer member and a sealing member by a sealing member or a columnar spacer in a form of a bank.
Further, Japanese Patent Application Laid-Open No. 2001-166121 discloses provision of a cutting mark as an indication of a position for cutting, near an intersection between cutting lines.
However, according to the method described in Japanese Patent Application Laid-Open No. 2003-215622, when using the sealing member to form an enclosure for enclosing the transfer member, there is still a concern that the sealing member as the bank itself may spread out, and as a result, it is not possible to provide the transfer member near a cutting line, and a contribution of this method to bezel slimming is not that much.
Further, according to the method described in International Publication No. WO00/45360, when using a columnar spacer to form a projection for enclosing both the sealing member and the transfer member, an enclosure in a form of the bank is to be additionally provided around the sealing member only to prevent spreading out. Therefore, a region for forming an enclosure in a form of a bank itself increases, and a contribution of this method to bezel slimming is limited.
Moreover, according to the method described in Japanese Patent Application Laid-Open No. 2006-268020, when forming the transfer member and an enclosure using a columnar spacer for enclosing the transfer member inwardly from the sealing member for enclosing liquid crystals, there is a possibility that the transfer member is brought into contact with the liquid crystals while it is possible to prevent an unnecessary region from increasing in the outside of the sealing member. This results in a concern of a reduced yield ratio brought by factors such as display defects due to contamination of the liquid crystals.
Furthermore, as described in Japanese Patent Application Laid-Open No. 2001-166121, it is commonly employed to provide the cutting mark near the intersection between the cutting lines, and such a cutting mark is also formed in the case of surface close attachment and bezel slimming. However, the cutting mark is primarily provided as the indication of the position for cutting, and may not particularly affect spreading out of paste materials as typically formed by a thin film of a metal, a resin or the like. This means that the cutting mark described in Japanese Patent Application Laid-Open No. 2001-166121 may not solve the above problems.
As described above, there has not been proposed an effective method which may allow manufacturing of liquid crystal display devices with a sufficiently slim bezel both by surface close attachment and at a high yield ratio.


---
abstract: 'Given an operator ideal $\cal I$, a Banach space $E$ has the $\cal I$-approximation property if operators on $E$ can be uniformly approximated on compact subsets of $E$ by operators belonging to $\cal I$. In this paper the $\cal I$-approximation property is studied in projective tensor products, spaces of linear functionals, spaces of homogeneous polynomials (in particular, spaces of linear operators), spaces of holomorphic functions and their preduals.'
author:
- 'Sonia Berrios and Geraldo Botelho[^1]'
title: 'Approximation properties determined by operator ideals'
---

Introduction
============

Given Banach spaces $E$ and $F$, by ${\cal L}(E;F)$ we denote the Banach space of all bounded linear operators from $E$ to $F$ endowed with the usual operator sup norm. The subspaces of ${\cal L}(E;F)$ formed by all finite rank, compact and weakly compact operators are denoted by ${\cal F}(E;F)$, ${\cal K}(E;F)$ and ${\cal W}(E;F)$, respectively. For a subset $S$ of ${\cal L}(E;F)$, the symbol $\overline{S}^{\tau_c}$ represents the closure of $S$ with respect to the compact-open topology $\tau_c$.\
It is well known that a Banach space $E$ has\
$\bullet$ the approximation property (in short, $E$ has AP) if ${\cal L}(E;E) = \overline{{\cal F}(E;E)}^{\tau_c}$,\
$\bullet$ the compact approximation property (in short, $E$ has CAP) if ${\cal L}(E;E) = \overline{{\cal K}(E;E)}^{\tau_c}$,\
$\bullet$ the weakly compact approximation property (in short, $E$ has WCAP) if ${\cal L}(E;E) = \overline{{\cal W}(E;E)}^{\tau_c}$.\
The AP is a classic in Banach space theory (see [@Casazza]) and is one of the main subjects of Grothendieck [@Grothendieck]. The CAP has been more studied in the last decades and recently (see, e.g. [@choi-kim; @choi-kim-outro; @Caliskan3; @Caliskan1]), but it goes back to Banach [@banach p.237]. The WCAP has been studied more recently (see [@Caliskan1; @Caliskan4]). Having in mind that ${\cal F}, {\cal K}$ and $\cal W$ are operator ideals, the properties above can be regarded as particular instances of the following general concept:

\[def\]Let $\cal I$ be an operator ideal. A Banach space $E$ is said to have the [$\cal I$-approximation property]{} (in short, $E$ has $\cal I$-AP) if ${\cal L}(E;E) = \overline{{\cal I}(E;E)}^{\tau_c}$.

Several variants of the approximation property have been studied recently (see, e.g, [@choi-kim; @choi-kim-lee; @dops; @delgadoJMAA; @lo; @oja2008; @sinha]), including ones closely related to the $\cal I$-AP [@llo; @lmo; @oja].\
The selection of operator ideals instead of other classes of linear operators related to $\cal F$, $\cal K$ and $\cal W$ is justified by the fact that even the most basic results depend on the ideal property (cf. Section 3).\
It is clear that if $E$ has AP then $E$ has $\cal I$-AP for every operator ideal $\cal I$. In particular, Banach spaces with Schauder basis (e.g., $\ell_p$, $1 \leq p < \infty$, and $c_0$) have $\cal I$-AP for every operator ideal $\cal I$.\
Let us stress that different ideals may give rise to different approximation properties: (i) Willis [@Willis] showed that there are spaces with CAP but not with AP; (ii) Szankowski [@szankowski] proved that for $1 \leq p < 2$, $\ell_p$ has a subspace $S_p$ without CAP, so $S_{\frac32}$ has WCAP but not CAP and $S_1$ has ${\cal CC} \cap {\cal C}_2$-AP but not CAP, where ${\cal CC}$ and ${\cal C}_2$ are the ideals of completely continuous and cotype 2 operators, respectively. The fact that different operator ideals usually give rise to different approximation properties justifies the study of the $\cal I$-AP for arbitrary operator ideals, which is the aim of this paper. In Example \[example\] we shall see that different ideals may generate the same approximation property.\
The study of the approximation property and its already studied variants is very rich and multifaceted, so the study of the $\cal I$-AP could follow several different trends. This means that, to study the $\cal I$-AP, choices have to be made. In this paper we have chosen to study the $\cal I$-AP in projective tensor products (Section 5) and in spaces of mappings between Banach spaces, namely, spaces of linear functionals (Section 4), spaces of homogeneous polynomials (Section 6) and spaces of holomorphic functions and their preduals (Section 7). Proposition \[certo\] fixes and generalizes a result of [@Caliskan4].\
The results we prove in the different sections of the paper seem - at first glance - to be completely disconnected. Connections of results from different sections are given in Section 7.

 Notation and preliminaries
===========================

When $F$ is the scalar field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, we shall write $E'$ instead of $\mathcal{L}(E;\mathbb{K})$. The [*compact-open topology or topology of compact convergence*]{} is the locally convex topology $\tau_c$ on $\mathcal{L}(E;F)$ which is generated by the seminorms of the form $$p_K(T)=\sup_{x\in K}\|T(x)\|,$$ where $K$ ranges over all compact subsets of $E$.\
Given a subset $S$ of $\mathcal{L}(E;F)$, $\overline{S}^{\tau_c} = \mathcal{L}(E;F)$ if and only if for every $T \in \mathcal{L}(E;F)$, every compact set $K\subseteq E$ and every $\varepsilon >0$, there is an operator $U\in S$ such that $\|T(x)-U(x)\|<\varepsilon$ for every $x\in K$.

An [*operator ideal*]{} $\mathcal{I}$ is a subclass of the class of all continuous linear operators between Banach spaces such that for all Banach spaces $E$ e $F$, the component $\mathcal{I}(E;F)=\mathcal{L}(E;F)\cap \mathcal{I}$ satisfy:\
(a) $\mathcal{I}(E;F)$ is a linear subspace of $\mathcal{L}(E;F)$ which contains the finite rank operators.\
(b) Ideal property: If $T\in \mathcal{L}(E;F)$, $R\in
\mathcal{I}(F;G)$ and $S\in \mathcal{L}(G;H)$, then the composition $S\circ R\circ T$ is in $\mathcal{I}(E;H)$.

By $id_E$ we mean the identity operator on the Banach space $E$. For a given operator ideal $\cal I$, by $\overline{\cal I}$ we mean the closure of $\cal I$, that is, $\overline{\cal I}(E;F) = \overline{{\cal I}(E;F)}$ for every Banach spaces $E$ and $F$. For the theory of operator ideals we refer to [@Pietsch; @klaus]. Here is a list of the operator ideals occurring in this paper:\
$\cal F$ = finite rank operators (the range is finite-dimensional),\
$\cal A := \overline{\cal F}$ = approximable operators,\
$\cal K$ = compact operators (bounded sets are mapped onto relatively compact sets),\
$\cal W$ = weakly compact operators (bounded sets are mapped onto relatively weakly compact sets),\
${\cal CC}$ = completely continuous operators (weakly convergent sequences are sent to norm convergent sequences),\
${\cal N}_p$ = $p$-nuclear operators,\
${\cal C}_p$ = cotype $p$ operators,\
${\cal T}_p$ = type $p$ operators,\
$\cal D$ = dualisable operators,\
$\cal S$ = separable operators (the range is separable),\
$\cal DP := \mathcal{W}^{-{\rm 1}} \circ {\cal CC}$ = Dunford-Pettis operators,\
$\cal J$ = integral operators,\
$\cal SN$ = strongly nuclear operators,\
$\cal SS$ = strictly singular operators (restrictions to infinite-dimensional subspaces are never isomorphisms),\
$\cal SC$ = strictly cosingular operators,\
$\Pi_p$ = absolutely $p$-summing operators,\
$\Pi_{r,p,q}$ = absolutely $(r,p,q)$-summing operators [@Pietsch 17.1],\
$\Gamma_p$ = $p$-factorable operators,\
$\cal KC$ = K-convex operators [@klaus 31.1],\
$\cal QN$ = quasinuclear operators [@klaus Ex. 9.13],\
${\cal L}_{\infty, q, \gamma}$ = Lorentz-Zigmund operators [@cobos],\
${\cal U}_p $ = operators having approximation numbers belonging to $\ell_p$ [@Pietsch 14.2.4],\
${\cal L}_{p,q}$ = $(p,q)$-factorable operators,\
${\cal K}_p$ = $p$-compact operators (bounded sets are mapped to relatively $p$-compact sets),\
${\cal QN}_p$ = quasi $p$-nuclear operators [@delgadostudia].\

Basic results
=============

The results of this section, except for some implications of Proposition \[charac\], are elementary enough to have their proofs omitted. Nevertheless, it is worth mentioning that the ideal property plays a crucial role in their (easy) proofs.

The following characterizations are simple but useful.

\[propos\] Given an operator ideal $\cal I$, the following are equivalent for a Banach space $E$:\
[(a)]{} $E$ has the $\cal I$-approximation property.\
[(b)]{} $E$ has the $\overline{\cal I}$-approximation property.\
[(c)]{} $id_E \in \overline{{\cal I}(E;E)}^{\tau_c}$.\
[(d)]{} For each compact set $K\subseteq
E$ and every $\varepsilon
>0$, there is an operator $T\in \mathcal{I}(E;E)$ such that $\|T(x)-x\|<\varepsilon$ for every $x\in K$.

Given operator ideals ${\cal I}_1$ and ${\cal I}_2$, we say that ${\cal I}_1$-AP = ${\cal I}_2$-AP if the Banach spaces having ${\cal I}_1$-AP are exactly the ones having ${\cal I}_2$-AP. The equivalence between (a) and (b) in Proposition \[propos\] says that ${\cal I}$-AP = $\overline{\cal I}$-AP for every operator ideal $\cal I$. In particular,

Let ${\cal I}_1$ and ${\cal I}_2$ be operator ideals. If $\overline{{\cal I}_1} = \overline{{\cal I}_2}$, then ${\cal I}_1$-AP = ${\cal I}_2$-AP.

\[example\] Since ${\cal F} \subseteq {\cal N}_p \subseteq \overline{{\cal F}} = {\cal A}$ [@jarchow Proposition 19.7.3], it holds that ${\cal N}_p$-AP = AP whereas ${\cal F} \neq {\cal N}_p \neq \overline{{\cal F}} = {\cal A}$.

Let us see a few more interesting conditions that are equivalent to the $\cal I$-AP:

\[charac\] Let $\mathcal{I}$ be an operator ideal. The following statements are equivalent for a Banach space $E$:\
[(a)]{} $E$ has the $\mathcal{I}$-approximation property.\
[(b)]{} For every Banach $F$, $\mathcal{L}(E;F)=\overline{\mathcal{I}(E;F)}^{\tau_c}$.\
[(c)]{} For every Banach $F$, $\mathcal{L}(F;E)=\overline{\mathcal{I}(F;E)}^{\tau_c}$.\
[(d)]{} $\sum_{n=1}^{\infty}x'_n(T(x_n))=0$ for every $T \in {\cal L}(E;E)$ whenever the sequences $(x_n)\subseteq E$ and $(x'_n)\subseteq E'$ are such that $\sum_{n=1}^{\infty}\|x'_n\|\|x_n\|<\infty$ and $\sum_{n=1}^{\infty}x'_n(T(x_n))=0$ for every $T\in \mathcal{I}(E;E)$.\
[(e)]{} $\sum_{n=1}^{\infty}x'_n(x_n)=0$ whenever the sequences $(x_n)\subseteq E$ and $(x'_n)\subseteq E'$ are such that $\sum_{n=1}^{\infty}\|x'_n\|\|x_n\|<\infty$ and $\sum_{n=1}^{\infty}x'_n(T(x_n))=0$ for every $T\in \mathcal{I}(E;E)$.

\(a) $\Longrightarrow$ (b) and (a) $\Longrightarrow$ (c) are straightforward. (b) $\Longrightarrow$ (a), (c) $\Longrightarrow $ (a) and (d) $\Longrightarrow $ (e) are obvious.

\(e) $\Longrightarrow$ (a) Let $\varphi \in ({\cal L}(E;E),{\tau_c})'$ be such that $\varphi(T) = 0$ for every $T \in {\cal I}(E;E)$. By Grothendieck’s description [@Grothendieck] of the functionals belonging to $({\cal L}(E;E),{\tau_c})'$ (proofs can be found in [@lt Proposition 1.e.3] and [@du Lemma VIII.3.3]), there are sequences $(x_n)\subseteq E$ and $(x'_n)\subseteq E'$ such that $\sum_{n=1}^{\infty}\|x'_n\|\|x_n\|<\infty$ and $\varphi(T) = \sum_{n=1}^{\infty}x'_n(T(x_n))$ for every $T\in \mathcal{L}(E;E)$. By (d) we have that $\varphi(id_E) = \sum_{n=1}^{\infty}x'_n(x_n)=0$. Hence $\varphi(id_E) =0$ for every functional $\varphi \in ({\cal L}(E;E),{\tau_c})'$ that vanishes on ${\cal I}(E;E)$. By the Hahn-Banach theorem (see, e.g., [@Megginson Corollary 2.2.20]) it follows that $id_E \in \overline{{\cal I}(E;E)}^{\tau_c}$.

\(a) $\Longrightarrow$ (d) Assume that (d) does not hold. In this case there are sequences $(x_n)\subseteq E$ and $(x'_n)\subseteq E'$ such that $\sum_{n=1}^{\infty}\|x'_n\|\|x_n\|<\infty$, $\sum_{n=1}^{\infty}x'_n(T(x_n))=0$ for every $T\in \mathcal{I}(E;E)$ and $\sum_{n=1}^{\infty}x'_n(U(x_n))\neq0$ for some $U\in \mathcal{L}(E;E)$. Defining $$\varphi \colon \mathcal{L}(E;E) \longrightarrow \mathbb{K}~,~ \varphi(T)=\sum_{n=1}^{\infty}x'_n(T(x_n)),$$ by the above mentioned Grothendieck’s description we know that $\varphi\in
(\mathcal{L}(E;E), \tau_c)'$. Thus $\varphi$ vanishes on $\mathcal{I}(E;E)$ but $\varphi(U) \neq 0$. Calling on Hahn-Banach once more we conclude that $U \notin \overline{\mathcal{I}(E;E)}^{\tau_c}$, which contradicts (a).

As expected, $\cal I$-AP is inherited by complemented subspaces and is stable under the formation of finite cartesian products:

\[complemented\] Let $\mathcal{I}$ be an operator ideal and $E$ be a Banach space with the $\mathcal{I}$-approximation property. Then every complemented subspace of $E$ has the $\mathcal{I}$-approximation property as well.

\[sum\] Let $\mathcal{I}$ be an operator ideal, $k \in \mathbb{N}$ and $E_1, \ldots, E_k$ be Banach spaces. Then the finite direct sum (or cartesian product) $E=\bigoplus_{n=1}^{k} E_{n}$ has the $\mathcal{I}$-approximation property if and only if $E_1, \ldots, E_n$ have the $\mathcal{I}$-approximation property.

Duality
=======

In this section we study the dual properties of the ${\cal I}$-approximation property. Given an operator ideal $\cal I$ and Banach spaces $E$ and $F$, define $$\mathcal{I}^{dual}(E;F)=\{S\in \mathcal{L}(E;F) \,\,{\rm such \, that~the~adjoint~operator}\,\, S'\in \mathcal{I}(F';E')\}.$$ It is well known that $\mathcal{I}^{dual}$ is an operator ideal. By $J_E$ we mean the canonical embedding from $E$ to $E''$.

\[1propdualIAP\] Let ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$ be operator ideals. If $E'$ has ${\mathcal{I}}_2$-AP, $F$ is reflexive and ${\mathcal{I}}_2(F';E')\subseteq
{\mathcal{I}}_1^{dual}(F';E')$, then $\mathcal{L}(E;F)=\overline{\mathcal{I}_1(E;F)}^{\tau_c}$.

Let $V\in
\mathcal{L}(E;F)$ and let $\varphi \in (\mathcal{L}(E;F), \tau_c)'$ be such that $\varphi(T)=0$ for every $T\in \mathcal{I}_1(E;F)$. It is enough to show that $\varphi(V)=0$, because in this case $V\in \overline{\mathcal{I}_1(E;F)}^{\tau_c}$ by [@Megginson Corollary 2.2.20]. Calling on Grothendieck’s description of $(\mathcal{L}(E;F), \tau_c)'$ once more, there are sequences $(x_n)\subseteq E$ and $(y'_n)\subseteq F'$ such that $\sum_{n=1}^{\infty}\|y'_n\|\|x_n\|<\infty$ and $\varphi(U)=\sum_{n=1}^{\infty}y'_n(U(x_n))$. Let $S\in \mathcal{I}_2(F';E')$. By assumption we have that $S'\in
\mathcal{I}_1(E'';F'')$. From the reflexivity of $F$ we may define $R:=(J_F)^{-1}\circ S'\circ J_E \in \mathcal{I}_1(E;F)$. For every $u\in F''$ and $v\in F'$, $$\langle u, v \rangle= \langle J_F((J_F)^{-1}(u)),v \rangle= \langle v, (J_F)^{-1}(u) \rangle.$$ Observe that $\phi(\cdot)= \sum_n J_E(x_n)(\cdot)y'_n \in
(\mathcal{L}(F';E'), \tau_c)'$ and $$\begin{aligned}
\phi(S)&=& \sum_{n=1}^{\infty}
J_E(x_n)(S)y'_n=\sum_{n=1}^{\infty}J_E(x_n)(S(y'_n))
=\sum_{n=1}^{\infty}\langle J_E(x_n),S(y'_n) \rangle\\
&=&\sum_{n=1}^{\infty} \langle S'\circ J_E(x_n),y'_n \rangle =
\sum_{n=1}^{\infty} \langle y'_n, (J_F)^{-1}\circ S'\circ J_E(x_n) \rangle\\
&=&\sum_{n=1}^{\infty} \langle y'_n,R(x_n) \rangle=
\sum_{n=1}^{\infty}y'_n(R(x_n))=0.\end{aligned}$$ So $\phi(S) = 0$ for every $S\in \mathcal{I}_2(F';E').$ Since $E'$ has ${\cal{I}}_2$-AP, $\mathcal{L}(F';E')=\overline{\mathcal{I}_2(F';E')}^{\tau_c}$ by Proposition \[charac\]. Therefore [@Megginson Corollary 2.2.20] yields $\phi(V')=0$. Thus $$\begin{aligned}
0&=& \phi(V')
=\sum_{n=1}^{\infty}J_E(x_n)(V'(y'_n))
=\sum_{n=1}^{\infty}\langle J_E(x_n),V'(y'_n) \rangle\\
&=&\sum_{n=1}^{\infty} \langle V'(y'_n),x_n\rangle =\sum_{n=1}^{\infty} \langle y'_n,V(x_n) \rangle= \varphi(V).\end{aligned}$$ The proof is complete.

\[propdualIAP\] Let ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$ be operator ideals such that either ${\mathcal{I}}_2\subseteq {\mathcal{I}}_1^{dual}$ or ${\mathcal{I}}_2^{dual}\subseteq {\mathcal{I}}_1$ and $E$ be a reflexive Banach space.\
[(a)]{} If $E'$ has ${\mathcal{I}}_2$-AP then $E$ has ${\mathcal{I}}_1$-AP.\
[(b)]{} If $E$ has ${\mathcal{I}}_2$-AP then $E'$ has ${\mathcal{I}}_1$-AP.

Assume that ${\mathcal{I}}_2^{dual}\subseteq {\mathcal{I}}_1$. Let $u \in {\cal I}_2(E';E')$. Since $E$ is reflexive, $((J_E)^{-1}
\circ u' \circ J_E)' = u \in {\cal I}_2(E';E')$, hence $(J_E)^{-1}
\circ u' \circ J_E \in \mathcal{I}_2^{dual}(E;E)$. By assumption we have $(J_E)^{-1} \circ u' \circ J_E \in \mathcal{I}_1(E;E)$. Then $u' = J \circ (J_E)^{-1} \circ u' \circ J_E \circ (J_E)^{-1} \in
\mathcal{I}_1(E;E)$ by the ideal property, that is, $ u \in
\mathcal{I}_1^{dual}(E';E')$. We have just proved that $\mathcal{I}_2(E';E')\subseteq \mathcal{I}_1^{dual}(E';E')$. Since this condition holds trivially if $\mathcal{I}_2\subseteq
\mathcal{I}_1^{dual}$, in both cases we have $\mathcal{I}_2(E';E')\subseteq
\mathcal{I}_1^{dual}(E';E')$.\
(a) Suppose that $E'$ has ${\cal{I}}_2$-AP. Since $E$ is reflexive, we have $\mathcal{L}(E;E)=\overline{\mathcal{I}_1(E;E)}^{\tau_c}$ from Proposition \[1propdualIAP\], so $E$ has ${\cal{I}}_1$-AP.

\(b) Suppose that $E$ has ${\cal{I}}_2$-AP. Since $E$ and $E''$ are isometrically isomorphic, it follows that $E''$ has ${\cal{I}}_2$-AP. Hence $E'$ has ${\cal{I}}_1$-AP by (a).

\[dualIAP\] Let $\mathcal{I}$ be an operator ideal such that either $\mathcal{I}\subseteq \mathcal{I}^{dual}$ or $\mathcal{I}^{dual}\subseteq \mathcal{I}$ and $E$ be a reflexive Banach space. Then $E'$ has the $\mathcal{I}$-approximation property if and only if $E$ has the $\mathcal{I}$-approximation property.

Given $1 \leq p < \infty$, $p^*$ stands for the conjugate of $p$, that is $\frac1p + \frac{1}{p^*}=1$. For the definition of the adjoint ideal ${\cal I}^*$ of the operator ideal $\cal I$, see, e.g., [@DJT p. 132].

Let us see that there is plenty of ideals satisfying the conditions of Theorem \[propdualIAP\] and Corollary \[dualIAP\].\
(i) ${\cal N}_1^{dual} \subseteq {\cal J}$ [@klaus Ex. 16.9], ${\cal SS}^{dual} \subseteq {\cal SC}$ and ${\cal SC}^{dual} \subseteq {\cal SS}$ [@DJP 1.18], $\Gamma_p^{dual} = \Gamma_{p^*}$ [@DJT p. 186], $\Pi_1^{dual} = \Gamma_1^*$ [@DJT Corollary 9.5], ${\cal T}_p \subseteq {\cal C}_{p^*}^{dual}$ and ${\cal C}_{p^*} \circ {\cal KC} \subseteq {\cal T}_p^{dual}$ for $1 < p \leq 2$ [@klaus 31.2], ${\cal N}_1^{dual} \subseteq {\cal QN}$ [@klaus Ex. 9.13(b)], $\Pi_{r,p,q}^{dual} = \Pi_{r,q,p}$ [@Pietsch Theorem 17.1.5], ${\cal L}_{p,q}^{dual} = {\cal L}_{q,p}$ [@cdr p. 68], ${\cal K}_p = {\cal QN}^{dual}_p$ [@delgadostudia].\
(ii) The following ideals are completely symmetric (that is $\mathcal{I}= \mathcal{I}^{dual}$): ${\cal F}, {\cal A}, {\cal K}, {\cal W}$ [@Pietsch Proposition 4.4.7], $\cal J$ [@klaus Corollary 10.2.2], $\cal SN$ [@jarchow Theorem 19.9.3], ${\cal U}_p$, $0<p<\infty$ [@Pietsch Theorem 14.2.5] and $\cal KC$ [@klaus 31.1].\
(iii) The following ideals satisfy $\mathcal{I}\subseteq \mathcal{I}^{dual}$: ${\cal N}_1$ [@klaus 9.9] and $\cal D$ [@Pietsch Proposition 4.4.10].\
(iv) The following ideals satisfy $\mathcal{I}^{dual}\subseteq \mathcal{I}$: $\cal S$ [@Pietsch Proposition 4.4.8] and ${\cal DP}$ [@DJP 1.15].

Our next aim is to show that the implication $E'$ has $\cal{I}-$AP $\Longrightarrow$ $E$ has $\cal{I}$-AP holds in some situations not covered by Corollary \[dualIAP\]. A couple of concepts defined in [@choi-kim-outro] are needed:

Let $E$ be a Banach space. Consider in $\mathcal{L}(E;E)$ the topology, called $\nu$, for which a net $(T_\alpha)$ in $\mathcal{L}(E;E)$ converges to $T \in \mathcal{L}(E;E)$ if and only if $$\sum_{n=1}^{\infty}x'_n(T_{\alpha}(x_n)) \longrightarrow \sum_{n=1}^{\infty}x'_n(T(x_n))$$ for every $(x_n)\subseteq E$ and $(x^{'}_n)\subseteq E^{'}$ satisfying $\sum_{n=1}^{\infty}\|x'_n\|\|x_n\|<\infty$. In this case we write $T_\alpha \stackrel{\nu}\longrightarrow T$. It is immediate that the $\tau_c$-topology is stronger than the $\nu$-topology on $\mathcal{L}(E;E)$.\
The [*$weak^{*}$-topology*]{} on $\mathcal{L}(E';E')$ is the topology for which a net $(T_\alpha)$ in $\mathcal{L}(E';E')$ converges to $T \in \mathcal{L}(E';E')$ if and only if $$\sum_{n=1}^{\infty}(T_{\alpha}(x'_n))x_n \longrightarrow \sum_{n=1}^{\infty}(T (x'_n))x_n$$ for every $(x_n)\subseteq E$ and $(x^{'}_n)\subseteq E^{'}$ satisfying $\sum_{n=1}^{\infty}\|x'_n\|\|x_n\|<\infty$. In this case we write $T_\alpha \stackrel{weak^{*}} \longrightarrow  T$.\
The topology $\nu$ is stronger than the $weak^{*}$-topology on $\mathcal{L}(E';E')$. Moreover, for $T$ and a net $(T_\alpha)$ in $\mathcal{L}(E;E)$, $$\label{equation}T_\alpha \stackrel{\nu} \longrightarrow  T \Longleftrightarrow T'_\alpha \stackrel{weak^{*}} \longrightarrow  T'.$$

Given a Banach space $E$, be $w^*$ we mean the ordinary weak\* topology on $E'$. For a given operator ideal $\cal I$, by ${\cal I}_{w^*}(E';E')$ we denote the set of all operators belonging to ${\cal I}(E';E')$ which are $w^*$-to-$w^*$ continuous. The dual space $E'$ is said to have the [*weak\* density*]{} for $\cal{I}$ (in short, $E$ has $\cal{I}$-W\*D) if $$\mathcal{I}(E';E')\subseteq
\overline{\mathcal{I}_{w^*}(E';E')}^{weak^*}.$$

There are nonreflexive dual Banach spaces having $\cal{I}$-W\*D for every operator ideal $\cal I$. In [@choi-kim-outro Proposition 2.7(a)] it is proved that $\ell_1$ has $\cal{K}$-W\*D. The only feature of compact operators used in the proof is the ideal property, so the same lines prove that $\ell_1 = (c_0)'$ is a nonreflexive dual Banach space having $\cal{I}$-W\*D for every operator ideal $\cal I$.

So, formally Corollary \[dualIAP\] does not apply to dual spaces having $\cal{I}$-W\*D. In this direction we have:

Let $E$ be a Banach space and let $\mathcal{I}$ be an operator ideal such that $ \mathcal{I}^{dual} \subseteq \mathcal{I}$. If $E'$ has $\cal{I}$-AP and $\cal{I}$-W\*D, then $E$ has $\cal{I}$-AP.

Let $(x_n)\subseteq E$ and $(x^{'}_n)\subseteq E^{'}$ be sequences such that $\sum_{n=1}^{\infty}\|x'_n\|\|x_n\|<\infty$ and $\sum_{n=1}^{\infty}x'_n(T(x_n))=0$ for every $T\in
\mathcal{I}(E;E)$. We know that $id_{E'}\in \overline{\mathcal{I}(E';E')}^{\tau_c}$ because $E'$ has $\cal{I}$-AP, and that $\mathcal{I}(E';E')\subseteq
\overline{\mathcal{I}_{w^*}(E';E')}^{weak^*}$ because $E$ has $\cal{I}-W^*D$. Thus $id_{E'}\in \overline{\mathcal{I}_{w^*}(E';E')}^{weak^*}$ and then there is a net $(S_{\alpha})\subseteq \mathcal{I}_{w^*}(E';E')$ such that $S_{\alpha} \stackrel{weak^*}\longrightarrow id_{E'}$. For each $\alpha$, since $S_\alpha$ is $w^*$-to-$w^*$ continuous, there is $T_\alpha \in {\cal L}(E;E)$ such that $T_\alpha ' = S_\alpha$ (see [@varios Ex.3.20]). We know that $S_\alpha \in \mathcal{I}(E';E')$, so the condition $ \mathcal{I}^{dual} \subseteq \mathcal{I}$ yields that $T_\alpha  \in {\cal I}(E;E)$ for every $\alpha$. From (\[equation\]) and $$T_{\alpha}' = S_\alpha
\stackrel{weak^*}\longrightarrow id_{E'} = (id_E)'$$ we get $T_{\alpha}
\stackrel{\nu}\longrightarrow id_{E}$. So, by the definition of the $\nu$-convergence, $$\sum_{n=1}^{\infty}x'_n(T_{\alpha}(x_n)) \longrightarrow
\sum_{n=1}^{\infty}x'_n(id_E(x_n)) = \sum_{n=1}^{\infty}x'_n(x_n) .$$ But $\sum_{n=1}^{\infty}x'_n(T_{\alpha}(x_n))=0$ for every $\alpha$ because each $T_\alpha  \in {\cal I}(E;E)$, therefore $\sum_{n=1}^{\infty}x'_n(x_n)=0$. By Proposition \[charac\] it follows that $E$ has $\cal{I}$-AP.

Tensor stability
================

In this section we study the stability of $\cal I$-AP under the formation of projective tensor products. By $E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n$ we mean the completed projective tensor product of $E_1, \ldots, E_n$ ($\hat\otimes_\pi^n E$ if $E = E_1 = \cdots = E_n$), and by $\hat\otimes_\pi^{n,s}E$ the completed $n$-fold symmetric projective tensor product of $E$.

Given continuous linear operators $u_j \colon E_j \longrightarrow F_j$, $j = 1, \ldots, n$, by $u_1 \otimes \cdots\otimes u_n$ we denote the (unique) continuous linear operator from $E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n$ to $F_1 \hat\otimes_\pi \cdots \hat\otimes_\pi F_n$ such that $$u_1 \otimes \cdots\otimes u_n(x_1 \otimes \cdots \otimes x_n) = u_1(x_1) \otimes \cdots \otimes u_n(x_n)$$ for every $x_1 \in E_1, \ldots, x_n \in E_n$.

The proof of the stability of the approximation property with respect to the formation of projective tensor products relies heavily on the fact that $u_1 \otimes \cdots\otimes u_n$ is a finite rank operator whenever $u_1, \ldots, u_n$ are finite rank operators. Let us see that this does not hold for arbitrary operator ideals:

The identity operator $id_{\ell_2}$ is weakly compact but $id_{\ell_2} \otimes id_{\ell_2} = id_{\ell_2 \hat\otimes_\pi \ell_2}$ is not weakly compact because $\ell_2 \hat\otimes_\pi \ell_2$ fails to be reflexive.

In order to settle this difficulty we need the following methods of generating ideals of multilinear mappings from operator ideals. By ${\cal L}(E_1, \ldots, E_n;F)$ we denote the space of continous $n$-linear mappings from $E_1 \times \cdots \times E_n$ to $F$ endowed with the usual sup norm.

Let $\mathcal{I}, \mathcal{I}_1, \ldots, \mathcal{I}_n $ be operator ideals.\
(a) (Factorization Method) A continuous $n$-linear mapping $A\in \mathcal{L}(E_1, \ldots, E_n;F)$ is said to be of type $\mathcal{L}[\mathcal{I}_1, \ldots, \mathcal{I}_n]$ if there are Banach spaces $G_1, \ldots, G_n$, linear operators $u_j\in
\mathcal{I}_j(E_j;G_j)$, $j = 1, \ldots, n$, and an $n$-linear mapping $B\in \mathcal{L}(G_1, \ldots, G_n;F)$ such that $A=B\circ (u_1, \ldots, u_n)$. In this case we write $A \in {\mathcal{L}[\mathcal{I}_1, \ldots, \mathcal{I}_n]}(E_1, \ldots, E_n;F)$. If $\mathcal{I} = \mathcal{I}_1 = \cdots = \mathcal{I}_n $ we simply write $\mathcal{L}[\mathcal{I}]$.\
(b) (Composition ideals) A continuous $n$-linear mapping $A\in \mathcal{L}(E_1, \ldots, E_n;F)$ belongs to $\mathcal{I}\circ \mathcal{P}$ if there are a Banach space $G$, an $n$-linear mapping $B\in \mathcal{L}(E_1, \ldots, E_n;G)$ and a linear operator $u\in \mathcal{I}(G;F)$ such that $A =u\circ B$. In this case we write $A\in
\mathcal{I}\circ\mathcal{L}(E_1, \ldots, E_n;F)$.\
For details and examples we refer to [@Botelho1; @Botelho0].

\[prodtens\] Let $\mathcal{I}, \mathcal{I}_1, \ldots, \mathcal{I}_n $ be operator ideals such that $\mathcal{L}[\mathcal{I}_1, \ldots, \mathcal{I}_n] \subseteq \mathcal{I}\circ\mathcal{L}$. If $E_1$ has $\mathcal{I}_1$-AP, $\ldots$, $E_n$ has $\mathcal{I}_n$-AP, then $E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n$ has $\cal I$-AP.

Let $K$ be a compact subset of $E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n$. By [@klaus Corollary 3.5.1] there are compact sets $K_1 \subseteq E_1, \ldots, K_n \subseteq E_n$ such that $K$ is contained in the closure of the absolutely convex hull of $K_1 \otimes \cdots \otimes K_n :=$ $ \{x_1 \otimes \cdots \otimes x_n : x_1 \in K_1, \ldots, x_n \in K_n\}$. Since compact sets are bounded there is $M > 0$ such that $\|x_j \| \leq M$ for every $x_j \in E_j$, $j = 1, \ldots, n$. Let $\varepsilon > 0$. As $E_1$ has $\mathcal{I}_1$-AP, there is an operator $u_1 \in {\cal I}_1(E_1;E_1)$ such that $\|u_1(x_1) - x_1\| < \frac{\varepsilon}{2nM^{n-1}}$ for every $x_1 \in K_1$. As $E_2$ has $\mathcal{I}_2$-AP, there is an operator $u_2 \in {\cal I}_2(E_2;E_2)$ such that $\|u_2(x_2) - x_2\| < \frac{\varepsilon}{2nM^{n-1}\|u_1\|}$ for every $x_2 \in K_2$. Repeating the procedure we obtain operators $u_j \in {\cal I}_j(E_j;E_j)$ such that $$\|u_j(x_j) - x_j\| < \frac{\varepsilon}{2nM^{n-1}\|u_1\|\cdots \|u_{j-1}\|}$$ for every $x_j \in K_j$, $j = 1, \ldots, n$. Consider the canonical $n$-linear mapping $\sigma_n \colon E_1 \otimes \cdots \times E_n \longrightarrow  E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n$ given by $\sigma_n(x_1 , \ldots ,x_n) = x_1 \otimes \cdots \otimes x_n$ and observe that $\sigma_n \circ (u_1, \ldots, u_n) \in {\mathcal{L}[\mathcal{I}_1, \ldots, \mathcal{I}_n]}(E_1, \ldots, E_n;E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n)$. By assumption we have $\sigma_n \circ (u_1, \ldots, u_n) \in \mathcal{I} \circ {\mathcal{L}}(E_1, \ldots, E_n;E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n)$. Calling $T$ the linearization of $\sigma_n \circ (u_1, \ldots, u_n)$, by [@Botelho0 Proposition 3.2(a)] we have that $T \in {\cal I}(E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n;E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n)$. For every $x_1 \in E_1, \ldots, x_n \in E_n$, $$\begin{aligned}
T(x_1 \otimes \cdots \otimes x_n) &=& \sigma_n \circ (u_1, \ldots, u_n)(x_1, \ldots, x_n)\\
&=& \sigma_n (u_1(x_1), \ldots, u_n(x_n))\\
&=& u_1(x_1) \otimes \cdots \otimes u_n(x_n)\\
&=&  u_1 \otimes \cdots \otimes u_n(x_1 \otimes \cdots \otimes x_n).\end{aligned}$$ As both $T$ and $u_1 \otimes \cdots \otimes u_n $ are linear it follows that $T = u_1 \otimes \cdots \otimes u_n $, hence $u_1 \otimes \cdots \otimes u_n \in {\cal I}(E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n;E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n)$. We shall denote the projective norm of a tensor $z \in E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n$ by $\|z\|$ instead of $\pi(z)$. Given $x_1 \in K_1, \ldots, x_n \in K_n$, $$\|u_1 \otimes \cdots \otimes u_n (x_1 \otimes \cdots \otimes x_n) - x_1 \otimes \cdots\otimes x_n \|= \|u_1(x_1) \otimes \cdots \otimes u_n (x_n) - x_1 \otimes \cdots\otimes x_n \|$$ $$= \|u_1(x_1) \otimes \cdots \otimes u_n (x_n) - \sum_{j=1}^{n-1} u_1(x_1) \otimes \cdots \otimes u_j(x_j) \otimes x_{j+1} \otimes \cdots \otimes x_n \hspace*{30em}$$ $$+ \sum_{j=1}^{n-1} u_1(x_1) \otimes \cdots \otimes u_j(x_j) \otimes x_{j+1} \otimes \cdots \otimes x_n   - x_1 \otimes \cdots\otimes x_n \| \hspace*{5em}$$ $$= \left\|\sum_{j=1}^n u_1(x_1)\otimes \cdots \otimes u_{j-1}(x_{j-1}) \otimes (u_j(x_j) - x_j)\otimes x_{j+1} \otimes \cdots \otimes x_n   \right\|\hspace*{30em}$$ $$\leq \sum_{j=1}^n \|u_1(x_1)\otimes \cdots \otimes u_{j-1}(x_{j-1}) \otimes (u_j(x_j) - x_j)\otimes x_{j+1} \otimes \cdots \otimes x_n \| \hspace*{30em}$$ $$\leq \sum_{j=1}^n \|u_1\|\|x_1\| \cdots \|u_{j-1}\|\|x_{j-1}\|\|u_j(x_j) - x_j\|\|x_{j+1}\|  \cdots \|x_n \| \hspace*{30em}$$ $$< \sum_{j=1}^n \|u_1\| \cdots \|u_{j-1}\|M^{n-1}\frac{\varepsilon}{2nM^{n-1}\|u_1\|\cdots \|u_{j-1}\|} = \frac{\varepsilon}{2}. \hspace*{30em}$$ In summary, $$\|u_1 \otimes \cdots \otimes u_n (x_1 \otimes \cdots \otimes x_n) - x_1 \otimes \cdots\otimes x_n \| < \frac{\varepsilon}{2},$$ for every $x_1 \in K_1, \ldots, x_n \in K_n$. Take $z$ in the absolutely convex hull of $K_1 \otimes \cdots \otimes K_n$. Then $z = \sum_{j=1}^k \lambda_j x_j^1 \otimes \cdots \otimes x_j^n$, where $k \in \mathbb{N}$, $\lambda_1,\ldots, \lambda_k$ are scalars with $|\lambda_1| + \cdots + |\lambda_k |\leq 1$, and $x_j^m \in K_m$ for $j = 1, \ldots, k, m = 1, \ldots, n$. Then $$\|u_1 \,\otimes \cdots \otimes u_n(z)~ - ~z\|  \hspace*{30em}$$ $$= \left\|u_1 \otimes \cdots \otimes u_n\left(\sum_{j=1}^k \lambda_j x_j^1 \otimes \cdots \otimes x_j^n \right)- \sum_{j=1}^k \lambda_j x_j^1 \otimes \cdots \otimes x_j^n  \right\| \hspace*{30em}$$ $$= \left\| \sum_{j=1}^k \lambda_j \left(u_1 \otimes \cdots \otimes u_n\left( x_j^1 \otimes \cdots \otimes x_j^n \right)- x_j^1 \otimes \cdots \otimes x_j^n \right)  \right\| \hspace*{30em}$$ $$\leq   \sum_{j=1}^k |\lambda_j|\left\|\left(u_1 \otimes \cdots \otimes u_n\left( x_j^1 \otimes \cdots \otimes x_j^n \right)- x_j^1 \otimes \cdots \otimes x_j^n \right)  \right\|   \hspace*{30em}$$ $$<  \frac{\varepsilon}{2}\sum_{j=1}^k |\lambda_j| = \frac{\varepsilon}{2}.\hspace*{30em}$$ By continuity we have that $$\|u_1 \otimes \cdots \otimes u_n(z) - z\|\leq \frac{\varepsilon}{2} < \varepsilon$$ for every $z$ in the closure of the absolutely convex hull of $K_1 \otimes \cdots \otimes K_n$, hence for every $z \in K$.

As to ideals satisfying the conditions above we have:

\(a) It is plain that $\mathcal{L}[\mathcal{F}] \subseteq \mathcal{F}\circ\mathcal{L}$ and $\mathcal{L}[\mathcal{S}] \subseteq \mathcal{S}\circ\mathcal{L}$.\
(b) $\mathcal{L}[\mathcal{N}_1] \subseteq \mathcal{N}_1\circ\mathcal{L}$ [@holub1 Theorem 3.7] (see also [@jarchow Proposition 17.3.9]).\
(c) $\mathcal{L}[\mathcal{J}] \subseteq \mathcal{J}\circ\mathcal{L}$ [@holub Theorem 2].\
(d) Let ${\cal L}_{\cal K}$ denote the ideal of compact multilinear mappings (bounded sets are sent to relatively compact sets). Pe[ł]{}czyński [@pel] proved that ${\cal K} \circ {\cal L} = {\cal L}_{\cal K}$. Now it follows easily that $\mathcal{L}[\mathcal{K}] \subseteq \mathcal{K}\circ\mathcal{L}$. So the projective tensor product of spaces with CAP has CAP too.\
(e) $\mathcal{L}[\mathcal{L}_{\infty, q, \gamma}] \subseteq \mathcal{L}_{\infty, q, \gamma}\circ\mathcal{L}$ for $0<q\leq1$ and $-1/q < \gamma < \infty$ [@cobos Theorem 3.1].\
(f) $\mathcal{L}[\mathcal{L}_{1,q}] \subseteq \mathcal{L}_{1, q}\circ\mathcal{L}$ for $q> 1$ and $\mathcal{L}[\mathcal{K}_{1,p}] \subseteq \mathcal{K}_{1, p}\circ\mathcal{L}$ for $p \geq 1$ [@cdr Theorem 2.1].\
(g) It is unknown if the projective tensor product of Schur spaces is a Schur space (see, e.g., [@pams]), so it is unknown if $\mathcal{L}[\mathcal{CC}] \subseteq \mathcal{CC}\circ\mathcal{L}$.

Here are other concrete situations to which Proposition \[prodtens\] applies:

\[racher\] Let $n \in \mathbb{N}$.\
[(a)]{} If $1 \leq p_1, \ldots, p_n < \infty$, then ${\cal L}[{\cal W}, {\cal I}_1, \ldots, {\cal I}_n] \subseteq {\cal W} \circ {\cal L}$ where ${\cal I}_j$ is either ${\cal K}$ or $\Pi_{p_j}$, $j = 1, \ldots, n$.\
[(b)]{} ${\cal L}[\Pi_1, {\cal J}, \stackrel{(n)}{\ldots}, {\cal J}] \subseteq \Pi_1 \circ {\cal L}$.\
[(c)]{} ${\cal L}[{\cal QN}, {\cal N}_1, \stackrel{(n)}{\ldots}, {\cal N}_1] \subseteq {\cal QN} \circ {\cal L}$.\
[(d)]{} If $p_1 > p_j$ for $j = 2, \ldots, n$, then ${\cal L}[{\cal U}_{p_1}, {\cal U}_{p_2}, \ldots, {\cal U}_{p_n}] \subseteq {\cal U}_{p_1} \circ {\cal L}$.

\(a) Given an $n$-linear mapping $A \in {\cal L}[{\cal W}, {\cal I}_1, \ldots, {\cal I}_n](E,E_1, \ldots, E_;F)$, write $A = B \circ (u,u_1, \ldots, u_n)$ with $u\in
\mathcal{W}(E;G)$, $u_j\in
\mathcal{I}_j(E_j;G_j)$, $j = 1, \ldots, n$, and $B\in \mathcal{L}(G, G_1, \ldots, G_n;F)$. Since $u$ is weakly compact and $u_1$ is either compact or absolutely $p_1$-summing, by a result of Racher [@racher] we have that $u \otimes u_1$ is weakly compact. As $u_2$ is either compact or absolutely $p_2$-summing and the projective tensor norm is associative, $u \otimes u_1 \otimes u_2 = (u \otimes u_1) \otimes u_2$ is weakly compact by the same result of [@racher]. Repeating this procedure finitely many times we conclude that $u \otimes u_1 \otimes \cdots \otimes u_n$ is weakly compact. Denoting by $\sigma_{n+1}\colon E \times E_1 \times \cdots \times E_n \longrightarrow E \hat\otimes_\pi E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n  $ the canonical $(n+1)$-linear mapping and by $T$ the linearization of $B$, we conclude that $A = T \circ (u \otimes u_1 \otimes \cdots \otimes u_n) \circ \sigma_{n+1}$. It follows that $A \in {\cal W} \circ {\cal L}(E,E_1, \ldots, E_;F)$ because $T \circ (u \otimes u_1 \otimes \cdots \otimes u_n)$ is weakly compact by the ideal property.

For (b), (c) and (d), repeat the proof above using, instead of Racher’s result, the following results: in (b) and (c), two results due to Holub [@holub]: (i) If $u_1 \in \Pi_1(E_1;F_1)$ and $u_2 \in {\cal J}(E_2;F_2)$ then $u_1 \otimes u_2 \in \Pi_1(E_1 \hat\otimes_\pi E_2; F_1 \hat\otimes_\pi F_2)$; (ii) If $u_1 \in {\cal QN}(E_1;F_1)$ and $u_2 \in {\cal N}_1(E_2;F_2)$ then $u_1 \otimes u_2 \in {\cal QN}(E_1 \hat\otimes_\pi E_2; F_1 \hat\otimes_\pi F_2)$. In (d), the following result, which appears in König [@konig p. 79] and is credited to Pietsch [@pietscharchiv]: if $u_1 \in {\cal U}_{p_1}(E_1;F_1)$, $u_2 \in {\cal U}_{p_2}(E_2;F_2)$ and $p_1 > p_2$, then $u_1 \otimes u_2 \in {\cal U}_{p_1}(E_1 \hat\otimes_\pi E_2; F_1 \hat\otimes_\pi F_2)$.

Combining Proposition \[prodtens\] and Lemma \[racher\] we get:

 \
[(a)]{} Let $E_1, \ldots, E_n$ be Banach spaces, one of which with WCAP and the others $E_j$ with either CAP or $\Pi_{p_j}$-AP for some $1 \leq p_j < \infty$. Then $E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n$ has WCAP.\
[(b)]{} Let $E_1, \ldots, E_n$ be Banach spaces, one of which with $\Pi_1$-AP and the others with $\cal J$-AP. Then $E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n$ has $\Pi_1$-AP.\
[(c)]{} Let $E_1, \ldots, E_n$ be Banach spaces, one of which with $\cal QN$-AP and the others with AP. Then $E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n$ has AP.\
[(d)]{} Let $0 < p_1, \ldots, p_n$. If $E_1, \ldots, E_n$ are Banach spaces, each $E_j$ with ${\cal U}_{p_j}$-AP, then $E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n$ has ${\cal U}_{p_k}$-AP if $p_k > p_j$ for every $j \neq k$.

Let $E$ be a Banach space with WCAP but not with CAP (see the Introduction), and $E_1, \ldots, E_n$ be Banach spaces such that each $E_j$ has either CAP or $\Pi_{p_j}$-AP, $1 \leq p_1, \ldots, p_n < \infty$. Then $E  \hat\otimes_\pi E_1 \hat\otimes_\pi \cdots \hat\otimes_\pi E_n$ has WCAP.

\[cortens\] Let $\cal I$ be an operator ideal such that $\mathcal{L}[\mathcal{I}] \subseteq \mathcal{I}\circ\mathcal{L}$. The following are equivalent for a Banach space $E$:\
[(a)]{} $E$ has $\cal I$-AP.\
$\hat\otimes_\pi^n E$ has $\cal I$-AP for every $n \in \mathbb{N}$.\
$\hat\otimes_\pi^n E$ has $\cal I$-AP for some $n \in \mathbb{N}$.\
$\hat\otimes_\pi^{n,s}E$ has $\cal I$-AP for every $n \in \mathbb{N}$.\
$\hat\otimes_\pi^{n,s}E$ has $\cal I$-AP for some $n \in \mathbb{N}$.

\(a) $\Longrightarrow$ (b) follows from Proposition \[prodtens\]; (b) $\Longrightarrow$ (c) is obvious; (c) $\Longrightarrow$ (a) follows from Proposition \[complemented\] because $E$ is obviously a complemented subspace of $\hat\otimes_\pi^n E$; (b) $\Longrightarrow$ (d) follows from Proposition \[complemented\] because $\hat\otimes_\pi^{n,s}E$ a complemented subspace of $\hat\otimes_\pi^n E$ via the symmetrization operator; (d) $\Longrightarrow$ (e) is obvious; (e) $\Longrightarrow$ (a) follows from Proposition \[complemented\] because $E$ is a complemented subspace of $\hat\otimes_\pi^{n,s}E$ (see [@blasco Corollary 4]).

Polynomial ideals and the $\mathcal{I}$-AP
==========================================

The symbol $\mathcal{P}(^n E;F)$ stands for the space of continuous $n$-homogeneous polynomials from $E$ to $F$. A [*polynomial ideal*]{} is a subclass $\cal Q$ of the class of all continuous homogeneous polynomials between Banach spaces such that, for every $n \in \mathbb{N}$ and Banach spaces $E$ and $F$, the component ${\cal Q}(^nE;F):=\mathcal{P}(^nE;F)\cap {\cal Q}$ satisfy:\
(a) ${\cal Q}(^n E;F)$ is a linear subspace of $\mathcal{P}(^n E;F)$ which contains the $n$-homogeneous polynomials of finite type,\
(b) If $T\in \mathcal{L}(G;E)$, $P\in
{\cal Q}(^nE;F)$ and $S\in \mathcal{L}(F;H)$, then $S\circ P\circ T \in {\cal Q}(^nG;H)$.

There are different ways to construct a polynomial ideal from a given operator ideal $\mathcal{I}$. Let us see three of such methods (see [@Botelho1; @Botelho0]):

Let $\mathcal{I}$ be an operator ideal.\
(a) (Factorization Method) A continuous $n$-homogeneous polynomial $P\in \mathcal{P}(^n E;F)$ is said to be of type $\mathcal{L}[\mathcal{I}]$ if there are a Banach space $G$, a linear operator $u\in
\mathcal{I}(E;G)$ and a polynomial $Q\in \mathcal{P}(^n G;F)$ such that $P=Q\circ u$. In this case we write $P \in {\cal P}_{\mathcal{L}[\mathcal{I}]}(^nE;F)$\
(b) (Composition ideals) A continuous $n$-homogeneous polynomial $P\in \mathcal{P}(^n E;F)$ belongs to $\mathcal{I}\circ \mathcal{P}$ if there are a Banach space $G$, a polynomial $Q\in \mathcal{P}(^n G;F)$ and a linear operator $u\in \mathcal{I}(E;G)$ such that $P=u\circ Q$. In this case we write $P\in
\mathcal{I}\circ\mathcal{P}(^n E;F)$.\
(c) (Linearization method) A continuous $n$-homogeneous polynomial $P\in \mathcal{P}(^n E;F)$ is said to be of type $[\mathcal{I}]$ if the linear operator $${\bar P}: E \rightarrow {\cal P}(^{n-1}E;F)~,~{\bar P}(x)(y) = {\check P}(x,y, \ldots, y)$$ belongs to ${\cal I}$. In this case we write $P \in {\cal P}_{[\mathcal{I}]}(^nE;F)$.\
It is well known that $\mathcal{L}[\mathcal{I}]$, $\mathcal{I}\circ \mathcal{P}$ and $[\mathcal{I}]$ are polynomial ideals.

Given a polynomial $P \in {\cal P}(^nE;F)$, by $\check P$ we mean the (unique) continuous symmetric $n$-linear mapping from $E^n$ to $F$ such that $P(x) = \check P(x, \ldots, x)$ for every $x \in E$.

\[polideal\] Let $\mathcal{I}$ be an operator ideal. The following are equivalent for a Banach space $E$:\
[(a)]{} $E$ has the $\mathcal{I}$-approximation property.\
[(b)]{} $\mathcal{P}(^nE;F
)=\overline{\mathcal{P}_{\mathcal{L}[\mathcal{I}]}(^n
E;F)}^{\tau_c}$ for every $n\in \mathbb{N}$ and every Banach space $F$.\
[(c)]{} $\mathcal{P}(^nE;F
)=\overline{\mathcal{P}_{\mathcal{L}[\mathcal{I}]}(^n
E;F)}^{\tau_c}$ for some $n\in \mathbb{N}$ and every Banach space $F$.\
[(d)]{} $\mathcal{P}(^nF;E)=\overline{\mathcal{I}\circ\mathcal{P}(^nF;E)}^{\tau_c}$ for every $n\in \mathbb{N}$ and every Banach space $F$. [(e)]{} $\mathcal{P}(^nF;E)=\overline{\mathcal{I}\circ\mathcal{P}(^nF;E)}^{\tau_c}$ for some $n\in \mathbb{N}$ and every Banach space $F$.\
Furthermore, if $\mathcal{L}[\mathcal{I}]
\subseteq \mathcal{I}\circ \mathcal{L}$, then the conditions above are also equivalent to:\
[(f)]{} $\mathcal{P}(^nE;F)=\overline{\mathcal{I}\circ\mathcal{P}(^nE;F)}^{\tau_c}$ for every $n\in \mathbb{N}$ and every Banach space $F$.\
[(g)]{} $\mathcal{P}(^nE;F)=\overline{\mathcal{I}\circ\mathcal{P}(^nE;F)}^{\tau_c}$ for some $n\in \mathbb{N}$ and every Banach space $F$.

\(a) $\Longrightarrow$ (b) Let $P\in \mathcal{P}(^n E;F)$, $K$ be a compact subset of $E$ and $\varepsilon >0$. Since $P$ is uniformly continuous on $K$, there is $\delta>0$ such that $\|P(y)-P(x)\|<\varepsilon $ whenever $\|y-x\|<\delta$, $x\in K$ and $y\in E$. By the ${\cal I}$-AP of $E$ there is an operator $T
\in \mathcal{I}(E;E)$ such that $\|T(x)-x\|< \delta $ for every $x\in K$. It follow that $\|P(T(x))-P(x)\|< \varepsilon $ for every $x\in K$. But $P\circ
T\in \mathcal{P}_{\mathcal{L}[\mathcal{I}]}(^n E;F)$, so we have that $P\in \overline{\mathcal{P}_{\mathcal{L}[\mathcal{I}]}(^n
E;F)}^{\tau_c}$.

\(c) $\Longrightarrow$ (a) Let $u\in \mathcal{L}(E;F)$, $K$ be a compact subset of $E$ and $\varepsilon >0$. Let $\varphi\in
E'$, $\varphi\neq 0$, and $a\in K$ be such that $\varphi(a)=1$. Define $P\in \mathcal{P}(^nE;F) $ by $P(x)=\varphi(x)^{n-1}u(x)$. Since $K_1:=\displaystyle\bigcup_{\varepsilon_i=\pm
1}(\varepsilon_1K+\varepsilon_2K+\cdots+\varepsilon_nK)$ is a compact subset of $E$, by assumption there is a polynomial $Q\in
\mathcal{P}_{\mathcal{L}[\mathcal{I}]}(^n E;F)$ such that $\|Q(x)-P(x)\|<\frac{n!\varepsilon}{n}$ for every $x\in K_1$. By the polarization formula, for every $(x_1, x_2, \ldots ,
x_n)\in K\times K\cdots \times K$ we have $$\|\check{Q}(x_1,
x_2, \ldots , x_n)-\check{P}(x_1, x_2, \ldots , x_n)  \| \hspace*{15em}$$ $$=
\left\|\dfrac{1}{n!2^n}\sum_{\varepsilon_i=\pm 1}
\varepsilon_1\varepsilon_2\cdots \varepsilon_n \left[Q\left(\sum_{i=1}^{n}\varepsilon_i
x_i\right)- P\left(\sum_{i=1}^{n}\varepsilon_i x_i\right)\right]\right\|
<\dfrac{\varepsilon}{n}.$$ From $$\check{P}(x, a, \ldots , a)=
\dfrac{1}{n}u(x)+\dfrac{(n-1)}{n}\varphi(x)u(a),$$ it results that $$\|n\check{Q}(x, a, \ldots , a)-u(x)-(n-1)\varphi(x)u(a) \|\hspace*{15em}$$ $$= n\left\|\check{Q}(x, a, \ldots ,
a)-\left(\dfrac{1}{n}u(x)+\dfrac{(n-1)}{n}\varphi(x)u(a)\right)\right\|
<\varepsilon$$ for every $x\in K$. Considering $S=n\check{Q}(\cdot, a, \ldots ,
a)-(n-1)\varphi(\cdot)u(a)\in \mathcal{L}(E;F)$, we have $\|S(x)-u(x) \| <\varepsilon$ for every $x\in K$. Let us check that $S\in \mathcal{I}(E;F)$. Indeed, as $Q\in
\mathcal{P}_{\mathcal{L}[\mathcal{I}]}(^n E;F)$, there are a Banach space $G$, an operator $v\in \mathcal{I}(E;G)$ and a polynomial $R\in \mathcal{P}(^n G;F)$ such that $Q=R\circ v$. Define $T\colon G\longrightarrow F$ by $T(y)=\check{R}(y, v(a), \ldots ,
v(a))$. Then $T\circ v
\in \mathcal{I}(E;F)$ and $$T\circ v(x) = T(v(x))=\check{R}(v(x), v(a), \ldots , v(a))=
 \check{Q}(x, a, \ldots , a),$$ for every $x \in E$, proving that $\check{Q}(\cdot, a, \ldots , a)\in \mathcal{I}(E;F)$. On the other hand, the operator $\varphi(\cdot)u(a)\in \mathcal{I}(E;F)$ as it is a finite rank operator. Thus $S\in \mathcal{I}(E;F)$ and $\mathcal{L}(E;F)=\overline{\mathcal{I}(E;F)}^{\tau_c}$. Calling on Proposition \[charac\] we have that $E$ has $\cal{I}$-AP.

\(a) $\Longrightarrow$ (d) Let $P\in \mathcal{P}(^n F;E)$, $K$ be a compact subset of $E$ and $\varepsilon >0$. Since $P(K)$ is a compact subset of $E$ and $E$ has the $\mathcal{I}$-approximation property, there is an operator $T\in \mathcal{I}(E;E)$ such that $\|T(z)-z\|< \varepsilon$ for every $z\in P(K)$. Hence $\|T(P(x))-P(x)\|< \varepsilon $ for every $x\in K$. Since $T\circ P\in \mathcal{I}\circ\mathcal{P}(^n
F;E)$ we have that $P\in \overline{\mathcal{I}\circ\mathcal{P}(^n
F;E)}^{\tau_c}$.

\(e) $\Longrightarrow$ (a) The same argument of (c) $\Longrightarrow$ (a), [*mutatis mutandis*]{}, works in this case. We just sketch the proof: given an operator $u\in \mathcal{L}(F;E)$, a compact set $K \subseteq F$ and $\varepsilon >0$, take $\varphi\in
F'$, $\varphi\neq 0$, and $a\in K$ such that $\varphi(a)=1$. Defining $P=\varphi(\cdot)^{n-1}u(\cdot)\in \mathcal{P}(^nF;E) $ and a compact subset $K_1$ of $F$ as before, by assumption there is a polynomial $Q\in \mathcal{I}\circ \mathcal{P}(^n
F;E)$ such that $\|Q(x)-P(x)\|<\frac{n!\varepsilon}{n}$ for every $x\in K_1$. Define $S=n\check{Q}(\cdot, a, \ldots ,
a)-(n-1)\varphi(\cdot)u(a) \in {\cal L}(F;E)$ and proceed exactly as above to get $\|S(x)-u(x) \| <\varepsilon$ for every $x\in K$. Write $Q=v\circ R$ with $v\in \mathcal{I}(G;E)$ and $R\in \mathcal{P}(^n F;G)$ and define $T \in {\cal L}(F;G)$ by $T(y)=\check{R}(y, a, \ldots ,
a)$. Thus $v\circ T = \check Q(\cdot, a, \ldots, a) \in \mathcal{I}(F;E)$ and this implies that $S\in \mathcal{I}(F;E)$.

Since (b) $\Longrightarrow$ (c) and (d) $\Longrightarrow$ (e) are obvious and (b) $\Longrightarrow$ (a) and (c) $\Longrightarrow$ (a) follow by taking $n = 1$ in Proposition \[charac\], the first part of the proof is complete.

Assume now that $\mathcal{L}[\mathcal{I}]
\subseteq \mathcal{I}\circ \mathcal{L}$.\
(a) $\Longrightarrow$ (f) $E$ has $\mathcal{I}$-AP by assumption. Let $n\in \mathbb{N}$, $P\in \mathcal{P}(^n
E;F)$, $K$ be a compact subset of $E$ and $\varepsilon >0$. Note that $P= P_L\circ \sigma_n$ where $\sigma_n\in
\mathcal{P}(^n E; \hat\otimes_\pi^{n,s}E)$ is the canonical $n$-homogeneous polynomial defined by $\sigma_n(x)=
x\otimes \cdots \otimes x$ and $P_L\in
\mathcal{L}(\hat\otimes_\pi^{n,s}E; F)$ is the linearization of $P$, that is $P_L(x\otimes \cdots \otimes x)=P(x)$. By Corollary \[cortens\] we have that $\hat\otimes_\pi^{n,s}E$ has $\mathcal{I}$-AP, hence $\mathcal{L}(\hat\otimes_\pi^{n,s}E;
F)=\overline{\mathcal{I}(\hat\otimes_\pi^{n,s}E; F)}^{\tau_c}$ by Proposition \[charac\]. So for the compact subset $\sigma(K)$ of $\hat\otimes_\pi^{n,s}E$ there is an operator $u\in \mathcal{I}(\hat\otimes_\pi^{n,s}E; F)$ such that $$\|u \circ \sigma_n(x) - P(x)\| = \|u(\sigma_{n}(x))-P_L(\sigma_n(x)) \|<\varepsilon$$ for every $x\in K$. Since $Q=u\circ \sigma_{n}\in \mathcal{I}\circ\mathcal{P}(^nE;F)$ we have that $P\in
\overline{\mathcal{I}\circ\mathcal{P}(^nE;F)}^{\tau_c} $.\
(f) $\Longrightarrow$ (g) is obvious and (g) $\Longrightarrow$ (a) follows from a repetition of the argument of the proofs of (c) $\Longrightarrow$ (a) and (e) $\Longrightarrow$ (a), therefore the proof is complete.

The spaces $\mathcal{P}_{\mathcal{L}[\mathcal{I}]}(^n
E;E)$ and $\mathcal{I}\circ\mathcal{P}(^nE;E)$ are often different. We have obtained situations where, even though different, their $\tau_c$-closures coincide:

\[coincidence\] Let $\mathcal{I}$ be an operator ideal.\
[(a)]{} If the Banach spaces $E$ and $F$ have the $\mathcal{I}$-approximation property, then $\overline{\mathcal{P}_{\mathcal{L}[\mathcal{I}]}(^n
E;F)}^{\tau_c}=\mathcal{P}(^nE;F
)=\overline{\mathcal{I}\circ\mathcal{P}(^n F;E)}^{\tau_c}$ for every $n\in \mathbb{N}$.\
[(b)]{} A Banach space $E$ has the $\mathcal{I}$-approximation property if and only if $\overline{\mathcal{P}_{\mathcal{L}[\mathcal{I}]}(^n
E;E)}^{\tau_c}=\mathcal{P}(^nE;E
)=\overline{\mathcal{I}\circ\mathcal{P}(^n E;E)}^{\tau_c}$ for every $n\in \mathbb{N}$.

It is not difficult to check that neither $\mathcal{P}_{\mathcal{L}[\mathcal{W}]}(^2 \ell_1; \ell_1) \subseteq \mathcal{W} \circ \mathcal{P} (^2 \ell_1; \ell_1)$ nor $\mathcal{W} \circ \mathcal{P} (^2 \ell_1; \ell_1) \subseteq  \mathcal{P}_{\mathcal{L}[\mathcal{W}]}(^2 \ell_1; \ell_1)$ (see [@Botelho1 Examples 27 and 28]). Nevertheless, by Corollary \[coincidence\](b) both subspaces are $\tau_c$-dense in $\mathcal{P}(^2 \ell_1;\ell_1)$ because $\ell_1$ has the approximation property (hence has the weakly compact approximation property).

The following result appears in [@Caliskan4]:

\[errado\][[@Caliskan4 Theorem 11]]{} The following are equivalent for a Banach space $E$:\
[(a)]{} $E$ has the weakly compact approximation property.\
[(b)]{} $\mathcal{P}(^nE;F
)=\overline{\mathcal{P}_{[\mathcal{W}]}(^n
E;F)}^{\tau_c}$ for every $n\in \mathbb{N}$ and every Banach space $F$.\
[(c)]{} $\mathcal{P}(^nE;F
)=\overline{\mathcal{P}_{[\mathcal{W}]}(^n
E;F)}^{\tau_c}$ for some $n\in \mathbb{N}$ and every Banach space $F$.

Unfortunately there is a gap in the proof of this theorem (see the MathSciNet review of this paper by C. Boyd). In this direction we have:

\[certo\] Let $\cal I$ be a closed injective operator ideal. The following are equivalent for a Banach space $E$:\
[(a)]{} $E$ has the $\cal I$-approximation property.\
[(b)]{} $\mathcal{P}(^nE;F
)=\overline{\mathcal{P}_{[\mathcal{I}]}(^n
E;F)}^{\tau_c}$ for every $n\in \mathbb{N}$ and every Banach space $F$.\
[(c)]{} $\mathcal{P}(^nE;F
)=\overline{\mathcal{P}_{[\mathcal{I}]}(^n
E;F)}^{\tau_c}$ for some $n\in \mathbb{N}$ and every Banach space $F$.

Just combine Theorem \[polideal\] with the fact that $[\mathcal{I}] = {\cal L}[\mathcal{I}]$ whenever the operator ideal $\cal I$ is closed and injective (see [@braunss-junek]).

Recalling that $\cal W$ is closed and injective, Proposition \[certo\] fixes Theorem \[errado\] and generalizes it to arbitrary closed injective operator ideals.

Spaces of holomorphic functions
===============================

The approximation property and its variants in spaces of holomorphic functions and their preduals have been largely investigated (see, e.g., [@as; @Rueda; @Caliskan3; @Caliskan1; @dm1; @dm2; @mu]). In this section we study the $\cal I$-approximation property in spaces of holomorphic functions of bounded type, spaces of weakly uniformly continuous holomorphic functions, spaces of bounded holomorphic functions and/or their preduals. For background on infinite-dimensional holomorphy we refer to [@Dineen; @Mujica]. An important issue of this section is the combination of results from different sections of the paper.

All spaces in this section are supposed to be complex.\
Spaces of holomorphic functions, spaces of bounded holomorphic functions and spaces of weakly uniformly continuous holomorphic functions, as well as their corresponding preduals, are locally convex spaces, so we have to say a few words about the definition of the $\cal I$-approximation property in the setting of locally convex spaces. The definition of operator ideals (on Banach spaces) can be naturally generalized to the concept of operator ideals on locally convex spaces (details can be found in [@Pietsch Chapter 29]). We say that an operator ideal $\cal U$ on locally convex spaces is an extension of an operator ideal $\cal I$ on Banach spaces if $\mathcal{U}(E;F)=\mathcal{I}(E;F)$ for all Banach spaces $E$ and $F$. There are several ways to extend an operator ideal on Banach spaces to an operator ideal on locally convex spaces (see [@Pietsch Section 29.5]). In this paper we shall work with the smallest of such natural extensions, which we describe next. Given an operator ideal $\cal I$ on Banach spaces, an operator $S\in \mathcal{L}(U;V)$ between locally convex spaces belongs to the [*inferior extension of $\mathcal{I}$*]{} if there exist Banach spaces $E$ and $F$ and operators $A\in \mathcal{L}(U, E)$, $T\in \mathcal{I}(E, F)$ and $Y\in \mathcal{L}(F, V)$ such that $S=Y\circ T\circ A$. In this case, for the sake of simplicity, we still write $S \in \mathcal{I}(U;V)$. Of course we can consider the compact-open topology on ${\cal L}(U;U)$ for a locally convex space $U$, so Definition \[def\] makes sense for an operator ideal $\cal I$ on Banach spaces and a locally convex space $U$, hence the $\cal I$-approximation property is well defined for locally convex spaces.\
Unless explicitly stated otherwise, an operator ideal means an operator ideal on Banach spaces and an statement like ${\cal I}_1 \subseteq {\cal I}_2$ means that ${\cal I}_1(E;F) \subseteq {\cal I}_2(E;F)$ for all Banach spaces $E$ and $F$.

\[remark\]It is easy to see that Propositions \[propos\], \[complemented\] and \[sum\] hold true in the realm of locally convex spaces. Of course, in condition (d) of Proposition \[propos\], $\|T(x)-x\|$ is replaced by $p(T(x) -x)$ where $p$ is an arbitrary continuous semi-norm. The proof of the locally convex version of Proposition \[propos\] follows the lines of the proof of [@kothe 43(1)].

A sequence $\{E_n\}_{n=1}^{\infty}$ of subspaces of a locally convex space $E$ is said to be a [*decomposition of $E$*]{} if any $x \in E$ can be written in a unique way as $x = \sum_{n=1}^\infty x_n$ with $x_n \in E_n$ for every $n$ and the projection $\sum_{n=1}^\infty x_n \mapsto \sum_{n=1}^m x_n$ is continuous for every $m \in \mathbb{N}$.\
Let $\mathcal{S}=\{(\alpha_n)_{n=1}^{\infty}: \alpha_n\in \mathbb{C}
\,\,\,\text{and}
\,\,\,\limsup_{n\rightarrow\infty}|\alpha_n|^{\frac{1}{n}}\leq 1
\} $. A decomposition $\{E_n\}_{n=1}^{\infty}$ of $E$ is an [*$\mathcal{S}$-absolute decomposition*]{} if

1.  $\sum_{n=1}^{\infty} x_n\in E, \, x_n\in E_n$ for all $n$ and $(\alpha_n)_{n=1}^{\infty}\in \mathcal{S}$ implies $\sum_{n=1}^{\infty}\alpha_nx_n\in E,$

2.  If $p$ is a continuous semi-norm on $E$ and $(\alpha_n)_{n=1}^{\infty}\in \mathcal{S}$ then $$p_{\alpha}\left(\sum_{n=1}^{\infty}x_n
    \right):=\sum_{n=1}^{\infty}|\alpha_n|p(x_n)$$ defines a continuous semi-norm on $E$. \[condicao2\]

Further details can be found in [@Dineen Section 3.3].

\[decomposition\] Let $\mathcal{I}$ be an operator ideal. If $\{E_n\}_{n=1}^{\infty}$ is an $\mathcal{S}$-absolute decomposition of the locally convex space $E$, then $E$ has the $\mathcal{I}$-approximation property if and only if each $E_n$ has the $\mathcal{I}$-approximation property.

An adaptation of the proof of [@Rueda Proposition 1] works in this case. We give the details for the sake of completeness. Suppose that each $E_n$ has the $\mathcal{I}$-approximation property. Let $K$ denote a compact subset of $E$, let $p$ denote an arbitrary continuous semi-norm on $E$ satisfying condition $(2)$ above with $\alpha_n=1$ for all $n$, and let $\varepsilon>0$ be arbitrary.

Define $$\pi_{n}\left(\sum_{m=1}^{\infty} x_m \right):=\sum_{m=1}^{n}x_m$$ for all $\sum_{m=1}^{\infty} x_m\in E$ and let $\pi^n=id_E-\pi_n$. By [@Dineen Lemma 3.33] there exists a positive integer $n_0$ such that $$\sup\{ p(\pi^{n_0}(x)): x\in K\}<\varepsilon.$$ Consider $F_{n_0}:=\pi_{n_0}(E)=\sum_{n=1}^{n_0}E_n$. By the locally convex version of Proposition \[sum\] (see Remark \[remark\]) $F_{n_0}$ has the $\mathcal{I}$-approximation property. Since $\pi_{n_0}(K)$ is a compact subset of $F_{n_0}$ there exists an operator $T\in\mathcal{I}(F_{n_0};F_{n_0})$ such that $$p(\pi_{n_0}(x)-T(\pi_{n_0}(x)))<\varepsilon$$ for every $x\in K$. Using the natural inclusion $i \colon F_{n_0}\hookrightarrow E$ we see that $R:=i\circ T\circ \pi_{n_0} \in \mathcal{I}(E;E)$. Hence $$\begin{aligned}
p(x-R(x))&\leq&p(x-\pi_{n_0}(x))+p(\pi_{n_0}(x)-T(\pi_{n_0}(x))\\
&=&p(\pi^{n_0}(x))+p(\pi_{n_0}(x)-T(\pi_{n_0}(x))<2\varepsilon\end{aligned}$$ for every $x\in K$. So $id_E \in \overline{{\cal I}(E;E)}^{\tau_c}$. It follows that $E$ has $\cal I$-AP by the locally convex version of Proposition \[propos\].\
Conversely, since each $E_n$ is a complemented subspace of $E$ and $E$ has the $\mathcal{I}$-approximation property, by the locally convex version of Proposition \[complemented\] it follows that each $E_n$ has $\mathcal{I}$-approximation property as well.

By $\mathcal{P}_{w}(^nE)$ we mean the closed subspace of $\mathcal{P}(^nE)$ of all continuous $n$-homogeneous polynomials that are weakly continuous on bounded sets. Let $U$ be an open subset of a Banach space $E$. A bounded subset $A$ of $U$ is [*$U$-bounded*]{} if there is a 0-neighborhood $V$ such that $A+V \subseteq U$. By $\mathcal{H}_b(U;F)$ we denote the space of holomorphic functions $f \colon U \longrightarrow F$, where $F$ is a Banach space, of [*bounded type*]{}, that is, $f$ is bounded on $U$-bounded sets. If $F = \mathbb{C}$ we simply write $\mathcal{H}_b(U)$. The symbol $\mathcal{H}_{wu}(U)$ stands for the space of all holomorphic functions $f \colon U \longrightarrow \mathbb{C}$ that are weakly uniformly continuous on $U$-bounded sets. When endowed with the topology of uniform convergence on $U$-bounded sets, both $\mathcal{H}_b(U;F)$ and $\mathcal{H}_{wu}(U)$ are locally convex spaces.

\[holomor\] Let $\mathcal{I}$ be an operator ideal, $U$ be a balanced open subset of the Banach space $E$ and $F$ be a Banach space.\
[(a)]{} $\mathcal{H}_b(U;F)$ has $\mathcal{I}$-$AP$ if and only if $\mathcal{P}(^n E; F)$ has $\mathcal{I}$-$AP$ for every $n\in \mathbb{N}$.\
[(b)]{} $\mathcal{H}_{wu}(U)$ has $\mathcal{I}$-$AP$ if and only if $\mathcal{P}_{w}(^nE)$ has $\mathcal{I}$-$AP$ for every $n\in \mathbb{N}$.

Just combine Lemma \[decomposition\] with the facts that $\{\mathcal{P}(^n E; F)\}_{n=1}^\infty$ is an $\mathcal{S}$-absolute decomposition of $\mathcal{H}_b(U;F)$ (this follows from an adaptation of the proof of [@Dineen Proposition 3.36]) and that $\{\mathcal{P}_w(^n E)\}_{n=1}^\infty$ is an $\mathcal{S}$-absolute decomposition of $\mathcal{H}_{wu}(U;F)$ (see the proof of [@Rueda Theorem 9]).

In the sequel some of our apparently disconnected results will be combined altogether. A Banach space $E$ is said to be [*polynomially reflexive*]{} if ${\cal P}(^nE)$ is reflexive for every $n\in \mathbb{N}$. For example, Tsirelson’s original space $T^*$ is polynomially reflexive [@aad].

\[unification\] Let $\cal I$ be an operator ideal such that $\mathcal{L}[\mathcal{I}] \subseteq \mathcal{I}\circ\mathcal{L}$ and ${\cal I} \subseteq {\cal I}^{dual}$ or ${\cal I}^{dual} \subseteq {\cal I}$. The following are equivalent for a polynomially reflexive Banach space $E$ and a balanced open subset $U$ of $E$:\
[(a)]{} $E$ has $\cal I$-AP.\
[(b)]{} ${\cal P}(^nE)$ has $\cal I$-AP for every $n\in \mathbb{N}$.\
[(c)]{} ${\cal P}(^nE)$ has $\cal I$-AP for some $n\in \mathbb{N}$.\
[(d)]{} $\mathcal{H}_b(U)$ has $\cal I$-AP.

\(a) $\Longrightarrow$ (b) Let $n \in \mathbb{N}$. By Corollary \[cortens\] we know that $\hat\otimes_\pi^{n,s}E$ has $\cal I$-AP. Since ${\cal P}(^nE)$ is isomorphic to $\left(\hat\otimes_\pi^{n,s}E\right)'$ and these spaces are reflexive, by Corollary \[dualIAP\] we have that ${\cal P}(^nE)$ has $\cal I$-AP.\
(b) $\Longrightarrow$ (c) This implication is obvious.\
(c) $\Longrightarrow$ (a) By [@as Proposition 5.3] it follows that $E'$ is isomorphic to a complemented subspace of ${\cal P}(^nE)$, so $E'$ has $\cal I$-AP by Proposition \[complemented\]. Then $E$ has $\cal I$-AP by Corollary \[dualIAP\].\
(d) $\Longleftrightarrow$ (b) This equivalence follows from Corollary \[holomor\](a).

To get another connection of the results from different sections we consider the predual of the space of holomorphic functions: given an open subset $U$ of a Banach space, Mazet [@mazet] proved the existence of a complete locally convex space $G(U)$ and of a canonical holomorphic function $\delta_U \colon U \longrightarrow G(U)$ such that for every Banach space $F$ and every holomorphic function from $U$ to $F$ there is a unique continuous linear operator $T_f$ from $G(U)$ to $F$ such that $f = T_f \circ \delta_U$.

\[predual\] Let $U$ be a balanced open subset of the Banach space $E$ and $\mathcal{I}$ be an operator ideal such that $\mathcal{L}[\mathcal{I}] \subseteq \mathcal{I}\circ\mathcal{L}$. Then $E$ has $\cal I$-AP if and only if $G(U)$ has $\cal I$-AP.

For every $n \in \mathbb{N}$ let $Q(^nE)$ be the space of all linear functionals $\varphi$ on ${\cal P}(^nE)$ such that the restriction of $\varphi$ to each locally bounded subset is ${\tau}_c$-continuous. By [@boyd Proposition 4] we have that $\{Q(^nE)\}_{n=1}^\infty$ is an $\cal S$-absolute decomposition of $G(U)$, so, by Lemma \[decomposition\], $G(U)$ has $\cal I$-AP if and only if $Q(^nE)$ has $\cal I$-AP for every $n$. But $Q(^nE)$ is isomorphic to $\hat\otimes_\pi^{n,s}E$ (see [@boyd p.223]), so by Corollary \[cortens\] we have that $G(U)$ has $\cal I$-AP if and only if $Q(^nE)$ has $\cal I$-AP for every $n$ if and only if $\hat\otimes_\pi^{n,s}E$ has $\cal I$-AP for every $n$ if and only if $E$ has $\cal I$-AP.

The results from Section 6 have not been combined with results from other sections yet. For results of Section 6 to come into play we investigate the $\cal I$-approximation property in the predual of the space ${\cal H}^\infty(U;F)$ of bounded holomorphic functions from an open subset $U$ of a Banach space $E$ to a Banach space $F$. ${\cal H}^\infty(U;F)$ is a Banach space with the sup norm. Let $U$ be an open subset of a Banach space $E$. Mujica [@Mujica1] proved the existence of a Banach space $G^{\infty}(U)$ and of a canonical bounded holomorphic mapping $\delta_{U}\in {\cal{H}}^{\infty}(U; G^{\infty}(U))$ with the following universal property: to every $f\in\mathcal{H}^{\infty}(U; F)$ corresponds a unique linear operator $T_f \in \mathcal{L}(G^{\infty}(U); F)$ such that $f=T_f\circ \delta_U$. He also introduced a very useful locally convex topology on ${\cal H}^\infty(U;F)$:

[[@Mujica1 Theorem 4.8]]{}\[teomujica\] Let $E$ and $F$ be Banach spaces, and let $U$ be an open subset of $E$. Let $\tau_{\gamma}$ denote the locally convex topology on ${\cal H}^{\infty}(U; F)$ generated by the seminorms of the form $$p(f)=\sup_{j} \alpha_j\|f(x_j)\|,$$ where $(x_j)$ varies over all sequences in $U$ and $(\alpha_j)$ varies over all sequences of positive real numbers tending to zero. Then the mapping $$f\in (\mathcal{H^{\infty}}(U; F), \tau_{\gamma}) \longrightarrow T_f \in (\mathcal{L}(G^{\infty}(U), \tau_c)$$ is a topological isomorphism.

We denote by $\mathcal{I}\circ \mathcal{H}^{\infty}(U; F)$ the collection of all $f \in \mathcal{H}^{\infty}(U;F)$ so that $f=u\circ g$, where $G$ is a Banach space, $g\in
\mathcal{H}^{\infty}(U;G)$ and $u\in \mathcal{I}(G;F)$. Next result extends [@Caliskan1 Theorem 5].

Let $\mathcal{I}$ be an operator ideal such that $\cal{L}[\cal{I}]\subseteq \cal{I}\circ
\cal{L}$. The following conditions are equivalent for a Banach space $E$ and an open subset $U$ of $E$:\
[(a)]{} $ E$ has $\mathcal{I}$-$AP$.\
[(b)]{} $\mathcal{H}^{\infty}(U;F)= \overline{\cal{I}\circ
\cal{H}^{\infty}(U; F)}^{\tau_{\gamma}}$ for every Banach space $F$.\
[(c)]{} $G^{\infty}(U)$ has $\cal{I}$-AP.

\(a) $\Longrightarrow$ (b) Let $f\in
\mathcal{H^{\infty}}(U;F)$. Let $p$ be a continuous semi-norm on $
(\mathcal{H^{\infty}}(U;F), \tau_{\gamma})$. By [@Mujica1 Proposition 5.2] there are homogeneous polynomials $P_j\in P(^jE;F)$, $j=0, 1, \ldots,n$, such that $p(P-f)<\frac{\varepsilon}{2}$ where $P = P_0 +P_1+ \cdots + P_n$. Since $E$ has $\cal{I}$-AP and $\cal{L}[\cal{I}]\subseteq \cal{I}\circ
\cal{L}$, it follows from Proposition \[polideal\] that $\mathcal{P}(^jE;F)=\overline{\mathcal{I}\circ\mathcal{P}(^jE;F)}^{\tau_c}$ for every $j\in \mathbb{N}$. On the other hand, by [@Mujica1 Proposition 4.9], $\tau_{\gamma}= \tau_{c}$ on $P\in P(^{j}E;F)$ for every $j\in \mathbb{N}$. So there are homogeneous polynomials $Q_j\in \mathcal{I}\circ
\mathcal{P}(^j E;F)$ such that $$p(Q_j-P_j)<\frac{\varepsilon}{2(n+1)}$$ for every $j=0, 1, \ldots n$. Putting $Q=Q_0+Q_1+ \cdots +Q_n$, mimicking an argument used in the proof of [@Botelho2 Theorem 2.2] one can easily prove that $Q$ is of the form $Q=u\circ R$ where $u\in {\cal I}(E;G)$, $G$ is a Banach space and $R$ is a finite sum of homogeneous polynomials from $G$ to $F$. Then the restriction of $Q$ to $U$, still denoted by $Q$, is a bounded holomorphic function, so $Q \in \mathcal{I}\circ \mathcal{H^{\infty}}(U; F)$. Since $$p(Q-P) = p\left(\sum_{j=0}^n Q_j - \sum_{j=0}^n P_j  \right) \leq \sum_{j=0}^n p(Q_j - P_j) <\frac{\varepsilon}{2},$$ if follows that $$p(Q-f)\leq
p(Q-P)+p(P-f)< \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon,$$ which proves (b).

\(b) $\Longrightarrow$ (c) By [@Mujica1 Theorem 2.1], $\delta_{U}\in
\mathcal{H}^{\infty}(U;G^{\infty}(U))$. Taking $F=G^{\infty}(U)$ in (b), we have that $\delta_{U}\in
\overline{\mathcal{H}^{\infty}(U;G^{\infty}(U))}^{\tau_{\gamma}}$. Hence there is a net $(f_\alpha)\subseteq
\mathcal{H}^{\infty}(U;G^{\infty}(U))$ such that $\displaystyle
f_{\alpha}\stackrel{\tau_{\gamma}}\longrightarrow \delta_{U}$. As to the corresponding net $(T_{f_\alpha})$ of linear operators, by Theorem \[teomujica\] we get that $$T_{f_\alpha} \stackrel{\tau_c}{\longrightarrow} T_{\delta_U} = id_{G^\infty(U)}.$$ But [@Botelho2 Proposition 4.2] gives that $(T_{f_\alpha})\subseteq \mathcal{I}(G^{\infty}(U);
G^{\infty}(U) )$. Therefore $id_{G^{\infty}(U)}\in
\overline{\mathcal{I}(G^{\infty}(U); G^{\infty}(U) )}^{\tau_c}$. By Proposition \[charac\] we have that $G^{\infty}(U)$ has $\mathcal{I}-$AP.

\(c) $\Longrightarrow$ (a) By [@Mujica1 Proposition 2.3], $E$ is topologically isomorphic to a complemented subspace of $G^{\infty}(U)$, which has $\mathcal{I}$-AP by assumption. It follows from Proposition \[complemented\] that $E$ has $\mathcal{I}-$AP.

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Faculdade de Matemática\
Universidade Federal de Uberlândia\
38.400-902 - Uberlândia - Brazil\
e-mails: soniles@famat.ufu.br, botelho@ufu.br

[^1]: Supported by CNPq Grant 306981/2008-4.2010 Mathematics Subject Classification: 46B28, 47B10, 46G20, 46G25.