Qiujia Li, Bo Li, Dongseong Hwang, Tara N. Sainath, Pedro M. Mengibar

# Modular Domain Adaptation for Conformer-Based Streaming ASR

###### Abstract

Speech data from different domains has distinct acoustic and linguistic
characteristics. It is common to train a single multidomain model such as a
Conformer transducer for speech recognition on a mixture of data from all
domains. However, changing data in one domain or adding a new domain would
require the multidomain model to be retrained. To this end, we propose a
framework called _modular domain adaptation_ (MDA) that enables a single model
to process multidomain data while keeping all parameters domain-specific,
_i.e._ , each parameter is only trained by data from one domain. On a
streaming Conformer transducer trained only on video caption data,
experimental results show that an MDA-based model can reach similar
performance as the multidomain model on other domains such as voice search and
dictation by adding per-domain adapters and per-domain feed-forward networks
in the Conformer encoder.

Index Terms: speech recognition, domain adaptation, Conformer transducer

##  1 Introduction

Automatic speech recognition (ASR) systems have been used across various
applications, such as video captioning [1], dictation [2], voice search [3,
4], voice assistant [5] and telephony [6, 7, 8]. Each domain has its own
acoustic and linguistic characteristics. For example, video captioning data
has diverse acoustic environments and speaking styles; a dictation utterance
is likely to have little background noise; voice search data tends to have
short queries with named entities.

The performance of ASR models deteriorates significantly when the model is
trained on a particular domain but evaluated on another [9]. Given sufficient
model capacity and training data, it is desirable to build a single ASR model
to serve all application domains. To ensure the model performs well for all
domains, one simple and effective approach is to mix all data during training
to obtain a multidomain model [10, 11]. Since all model parameters are shared
across all domains, the multidomain approach has some shortcomings. First, the
entire model needs to be retrained if the training data from a certain domain
changes or a new domain is added. Second, finding the right balance to mix
data from various domains is nontrivial. On the other end of the spectrum, it
is also possible to build an ASR model for each domain. This means all
parameters are domain-specific where each parameter is trained on data from a
single domain. However, this is very expensive for training and serving as
multiple models need to be maintained. Therefore, it is ideal to build a
single model that serves all domains while keeping all model parameters
domain-specific.

The contribution of this work is as follows. First, we proposed a framework
called _modular domain adaptation_ (MDA) that can process multidomain data
with all parameters being domain-specific. As illustrated in Fig. 1, the
backbone model is trained on a particular domain (in red), and all traffic
passes through the common backbone model except certain parts of the model
where the traffic is split and routed through the per-domain parameters.
Second, we identified the most effective components of the Conformer backbone
model to integrate per-domain parameters for parameter efficiency. Also, two
types of per-domain parameters are explored, _i.e._ , per-domain components
that replace the existing components in the backbone model and per-domain
adapters that add lightweight adapter modules to modify intermediate
representations. Third, the final recipe was validated on three different
domains with a large amount of data and minimum word error rate (MWER)
training [12]. The MDA-based model achieved similar performance as the
multidomain model across all domains with 0.2–0.4% absolute degradation in
word error rates (WERs). The number of per-domain parameters is 22% of the
backbone model.

Figure 1: Modular domain adaptation (MDA). Different colors indicate
parameters trained on different domains. In this example, speech from domain 2
passes through the backbone trained on domain 1 except when per-domain
parameters are available.

Although domain adaptation has been widely studied for ASR [9] with various
paradigms including input feature adaptation [13, 14, 15], model-based
adaptation [16, 17, 18, 19, 20] and multi-task learning [21, 22], our work
pays particular attention to modularity. Unlike general adaptation techniques,
MDA satisfies the constraint that all model parameters are domain-specific.
This brings many advantages of modularity [23]: similar functions of the ASR
model are encoded with the same module while allocating distinct functions to
per-domain parameters; per-domain parameters can be constructed separately and
updated locally; parameter efficiency is much higher than finetuning the
entire model or have multiple single-domain models.

In this paper, two methods under the framework of MDA are described in Sec. 2.
Extensive experimental studies are then conducted in Sec. 3 & 4, which offer
insights into the Conformer transducers, especially in terms of modularity.
Conclusions are drawn in Sec. 5.

##  2 Methods

In this section, two types of methods for MDA are introduced. Both approaches
enable the inputs to be transformed differently based on their corresponding
domains. Per-domain components replace a particular part of the backbone model
with separate parameters for each domain. Per-domain adapters are additional
domain-specific modules added to the backbone model.

Figure 2: Conformer transducer with cascaded encoders.

###  2.1 Per-Domain Components

As illustrated in Fig. 2, a Conformer transducer [24] has an encoder with a
stack of Conformer blocks and a decoder containing a prediction network and a
joint network. A Conformer block contains a sequence of four modules - a feed-
forward network (FFN), a multi-head self-attention (MHSA) module, a
convolution module (Conv), and another feed-forward network. Each module in
the Conformer block has a residual connection, which is omitted in the figure
for simplicity. To enable processing streaming audio, a limited amount of
future context is available for MHSA and Conv modules [25].

To perform MDA, some components of the Conformer transducer can be customized
for each domain, _i.e._ , same architecture but with different parameters for
each domain. As shown in Fig. 3(a), the input examples are routed through
different components corresponding to their domains in different colors. The
granularity of a ``component'' can vary greatly. A component can be a module
in a Conformer block, an entire Conformer block, or even the entire encoder.
At one extreme, it is equivalent to having a model for each domain if all
components of the model are separate for each domain. Since some components
will have a greater impact on the model quality than others, the effect of
per-domain Conformer components will be examined in Sec. 4.

###  2.2 Per-Domain Adapters

Another approach to MDA is to have per-domain adapters in the Conformer model.
Adapters are commonly used as efficient modules to adapt large foundation
models for particular tasks without finetuning all model parameters [26, 27,
28, 29]. For MDA, per-domain adapters can be used to augment the hidden
representations of the backbone model accordingly. For each domain, the
adapter first uses a weight matrix
$\mathbf{W}_{\text{down}}\in\mathbb{R}^{d\times b}$ to project the input
representation $\mathbf{h}\in\mathbb{R}^{1\times d}$ to a lower-dimensional
space with bottleneck dimension $b$, followed by a nonlinear activation
function $f(\cdot)$, and then projects up to the original dimension using
$\mathbf{W}_{\text{up}}\in\mathbb{R}^{b\times d}$. A residual connection is
also applied around the adapter:

$\mathbf{h}\leftarrow\mathbf{h}+f(\mathbf{h}\mathbf{W}_{\text{down}})\mathbf{W}_{\text{up}}.\vspace{-0.25em}$
(1)

There are different ways to configure adapters for the backbone model. The
per-domain adapters can be either inserted in between two adjacent modules of
the model, _i.e._ , sequential adapters as shown in Fig. 3(b), or positioned
in parallel to some components of the backbone model, _i.e._ , parallel
adapters as shown in Fig. 3(c). Experimental results for transfer learning
have shown parallel adapters outperform sequential ones and modifying FFN
representations is more effective than other modules in Transformer models
[28]. In Sec. 4, experiments will be carried out on Conformers for MDA.

The aforementioned two approaches have their respective advantages. Per-domain
components do not increase the computation cost because certain components of
the backbone model are simply replaced. However, per-domain adapters have a
small increase in computation cost as extra parameters are inserted into the
backbone model. On the other hand, per-domain components generally need to
store a much larger number of domain-specific parameters than per-domain
adapters as a Conformer component is generally much larger than an adapter.

(a) Per-domain
components.

(b) Per-domain
sequential adapters.

(c) Per-domain
parallel adapters.

Figure 3: Various modular domain adaptation (MDA) methods.

##  3 Experimental Setup

###  3.1 Data

The training set contains three domains: YouTube (YT), voice search (VS) and
dictation (DT). In total, there are 755M utterances or 1.1M hours. For YT, the
training set has around 220M utterances or 600k hours, and the test set has
20k utterances or 380 hours. For VS, the training set has 520M utterances or
490k hours, and the test set has 9.4k utterances or 12 hours. For DT, the
training set has 10M utterances or 23k hours, and the test set has 17k
utterances or 39 hours. Utterances from all domains were anonymized and hand-
transcribed, except for the YT training set where the transcriptions were
obtained in a semi-supervised fashion [30]. Multi-condition training [31],
random data down-sampling to 8 kHz [32] and SpecAugment [33] were used during
training to increase data diversity. 128-dimensional filterbank features were
used as model inputs.

###  3.2 Model

The backbone model is a Conformer transducer model [24]. The encoder is based
on the cascaded structure [34, 35] which consists of 7 causal Conformer blocks
without future context followed by 10 non-causal Conformer blocks that use 900
ms of future acoustic context. The first two causal blocks do not have MHSA
modules. The model dimensions are 512 and 640 for causal and non-causal
encoders. For all Conformer blocks, MHSA modules have 8 attention heads; Conv
modules have a convolution kernel size of 15; and the dimension of the FFN
module is four times the model dimension. The causal and non-causal encoders
have 47M and 99M parameters respectively. The outputs of both encoders are
projected to 384 dimensions. There are two separate hybrid autoregressive
transducer (HAT) decoders [36] for causal and non-causal encoders. The output
vocabulary has 4096 wordpieces [37]. The joint network that combines features
from the encoder and the embedding prediction network [38] has 640 units. Each
decoder has 9.5M parameters. The total number of parameters of the entire
model is 165M. During training, two encoders are selected randomly with the
same probability and FastEmit loss [39] is applied with a scale of 0.005. All
models were trained using the Lingvo toolkit [40] for 300k steps on $8\times
8$ tensor processing units (TPUs) with a batch size of 4096.

##  4 Experimental Results

In this section, various MDA methods will be explored and compared with the
multidomain baseline. Note that all WERs reported are from the non-causal
decoder without using the end pointer (EP) [41] unless specified otherwise.

###  4.1 Baseline Conformer Model

The multidomain (MD) model is trained on a mixture of data from all five
domains. Only for the MD baseline model, the domain ID encoded by a
16-dimensional one-hot vector is appended to the acoustic features as input to
the model.

Table 1: WERs (%) of the multidomain and single-domain models. YT-only model will be used as the backbone model. | YT | VS | DT
---|---|---|---
multidomain (MD) | 14.0 | 4.7 | 3.4
YT-only | 13.9 | 25.1 | 8.3
VS-only | 43.0 | 4.5 | 14.7
DT-only | 37.2 | 7.4 | 3.7

As shown in Table 1, the single-domain model performs well on in-domain
testsets but struggles on out-of-domain testsets. The performance of the MD
model is about the same as the single-domain models on their respective
domains for YT and VS as they both have a large amount of training data. The
MD model has a lower WER than the DT-only model on the DT testset because the
size of the DT training set is an order of magnitude smaller than YT or VS.
Since YT is considered to be the most diverse domain containing speech in a
wide range of forms and acoustic conditions, the YT-only model is used as the
backbone model in the rest of the paper. Note that the parameters of the
backbone model will not be updated and per-domain parameters will be
initialized randomly from scratch.

###  4.2 Per-Domain Components

In this section, various components in the backbone YT-only model are replaced
by per-domain ones. For simplicity, only results on the VS testset are
reported and the observations on the DT domain are similar.

First, per-domain Conformer encoder blocks are examined. In Fig. 4, each bar
shows the WER of the model where a particular Conformer block is initialized
from scratch and trained on VS while the rest of the backbone model is frozen.
Fig. 4(a) shows that having per-domain Conformer blocks in the causal encoder
is not effective as the WER is far from the multidomain baseline (4.7%). The
trend line shows an interesting behavior where earlier blocks result in high
deletion error rates but later blocks cause high insertion error rates. This
may be attributed to the domain characteristics. The backbone YT-only model is
trained to transcribe not only the primary speakers, but also speakers in the
background. For VS, only the speech from the primary speaker is supposed to be
recognized. Additionally, the average duration of a YT training utterance is
3–4 times greater than VS. We hypothesize that the earlier blocks of the
encoder learn to suppress or enhance background signals. By training the
earlier blocks on VS, the model tends to drop the acoustic content too
aggressively whereas the later blocks cannot prevent the model to transcribe
all speech content. However, the picture changes greatly for per-domain
Conformer blocks in the non-causal encoder. As shown in Fig. 4(b), with the
help of future context, the WERs drop to around 6% for all non-causal
Conformer blocks. Further experiments show that having two per-domain
Conformer blocks yields marginal improvement over a single one.

(a) Causal encoder.

(b) Non-causal encoder.

Figure 4: WERs on VS testset using per-domain Conformer blocks. Note that the
scales of the y-axis are different. A causal / non-causal Conformer block has
6M / 10M parameters.

Instead of using per-domain Conformer blocks, it is also possible to have per-
domain Conformer modules across multiple Conformer blocks in the encoder.
Table 2 shows the WERs when using different domain-specific modules, which are
initialized from scratch and then trained on VS while the rest of the backbone
YT-only model remains fixed. For each Conformer module, two WER results are
presented where the ``NC'' column means only the non-causal encoder has per-
domain modules and ``C+NC'' means both encoders have per-domain modules. Three
conclusions can be drawn from the results in Table 2. First, by comparing the
last row, ``all params'', with the MD baseline, most modeling power of the
model resides in the encoders. Second, by comparing different Conformer
modules in the ``C+NC'' column, the majority of the modeling capability is
within the FFN start or FFN end module. Third, by comparing the ``NC'' and
``C+NC'' columns, non-causal blocks hold much more significance than causal
blocks in terms of the recognition quality on the target domain. Moreover,
compared with WERs shown in Fig. 4, having per-domain modules across all
blocks is more effective than having a per-domain Conformer block.

Table 2: WERs on VS testset using different per-domain encoder components. ``C / NC'' mean ``causal / non-causal'' encoders. | # trainable params | WER (%)
---|---|---
| C / NC (M) | NC | C+NC
MD baseline |  | 4.7
FFN start | 16.8 / 32.8 | 4.9 | 4.7
MHSA |  4.2 / 20.5 | 5.5 | 5.3
Conv |  5.0 / 12.4 | 5.4 | 5.0
FFN end | 16.8 / 32.8 | 5.0 | 4.7
all params | 46.8 / 99.1 | 4.8 | 4.6

In Table 3, per-domain decoder components are also tested. Having a per-domain
prediction network is not useful as it has a similar WER as the backbone YT-
only model (25.1%). This is expected because the prediction network only
provides the token embeddings of the two previous tokens for the joint network
[38]. The joint network plays the most important role within the decoder as it
transforms the encoder features, but the WER of the per-domain joint network
is far from the MD baseline. Further experiments have also been carried out to
combine domain-specific Conformer modules and per-domain joint networks, the
improvement over the results in Table 2 is small, which further validates the
finding that the main modeling capacity is within the encoder.

Table 3: WERs on VS testset using different per-domain decoder components. | # trainable params (M) | WER (%)
---|---|---
MD baseline |  | 4.7
prediction network | 6.1 | 27.0
joint network | 3.3 | 9.2
all params | 9.3 | 9.3

###  4.3 Per-Domain Adapters

Using per-domain adapters is a more parameter-efficient approach for MDA as
the number of additional domain-specific parameters is very small. Table 4
shows the WERs on the VS testset for both sequential adapters as in Fig. 3(b)
and parallel adapters as in Fig. 3(c). Since the FFN module is the most
important module in the Conformer as found in Sec. 4.2, both types of per-
domain adapters are applied around all the FFN (start & end) modules in the
encoder. There are three findings from Table 4. First, by comparing sequential
and parallel adapters of the same bottleneck dimension, parallel adapters
consistently outperform sequential ones. This is consistent with the
literature in other fields [28]. Second, a larger bottleneck dimension helps
for both types of adapters. Third, similar to Table 2, the adapters in the
non-causal encoder are very significant and the adapters in the causal encoder
bring small but consistent gains. Compared with the per-domain Conformer
blocks in Sec. 4.2, per-domain adapters use much fewer domain-specific
parameters while achieving lower WERs.

Table 4: WERs on VS testset using various per-domain adapters. ``C / NC'' mean ``causal / non-causal'' encoders. position | dim | # trainable params | WER (%)
---|---|---|---
C / NC (M) | NC | C+NC
MD baseline |  |  | 4.7
sequential | 64 | 1.1 / 1.7 | 6.2 | 5.9
128 | 2.1 / 3.3 | 5.9 | 5.7
256 | 4.2 / 6.6 | 5.7 | 5.4
parallel | 64 | 1.1 / 1.7 | 6.0 | 5.6
128 | 2.1 / 3.3 | 5.8 | 5.4
256 | 4.2 / 6.6 | 5.5 | 5.1

###  4.4 Final Recipe

Given that FFN modules have the most impact on WERs in the target domain and
domain adapters are very parameter efficient, together with the observation
that adapting the non-causal encoder is more effective than the causal
encoder, we can take advantage of both MDA approaches. Table 5 shows the WERs
on the VS testset with and without end pointing for three different setups.
Depending on the budget of per-domain parameters and the tolerance of quality
degradation compared to the MD baseline, a trade-off can be determined. As our
final recipe, per-domain adapters are added to the causal encoder to ensure
reasonable end-pointing latency [41, 35] by the causal encoder and per-domain
FFN modules are used in the non-causal encoder to improve the recognition
quality of the target domains.

Table 5: WERs on VS testset by various MDA methods with and without end pointing. PA refers to per-domain parallel adapters and FFN refers to per-domain FFN end. C | NC | # trainable params | WER (%)
---|---|---|---
C / NC (M) | w/o EP | w/ EP
MD baseline | 4.7 | 6.1
PA | PA |  4.2 / 6.6 | 5.1 | 6.6
PA | FFN |  4.2 / 32.8 | 4.7 | 6.4
FFN | FFN | 16.8 / 32.8 | 4.7 | 6.3

Finally, as shown in Table 6, the recipe is evaluated on all three domains and
MWER training [12] is applied afterwards. During the MWER training, all model
parameters are updated for the MD baseline, but only per-domain parameters are
updated for the MDA recipe. The total number of per-domain parameters is
around 37M and the WER degradation compared to the MD baseline after MWER
training is 0.2–0.4% absolute.

Table 6: WERs of the final recipe on all three domains and the effect of the MWER training. WERs on VS are with EP. | MWER | YT | VS | DT
---|---|---|---|---
MD baseline | ✗ | 14.0 | 6.1 | 3.4
✓ | 13.5 | 5.7 | 3.1
MDA recipe | ✗ | 14.1 | 6.4 | 3.7
✓ | 13.7 | 5.9 | 3.5

##  5 Conclusions

For Conformer models, modular domain adaptation (MDA) approaches such as per-
domain feed-forward networks and per-domain adapters can achieve similar
recognition quality as a multidomain model while having the benefits of
modularity by using all domain-specific parameters for the entire model. In
the future, we plan to explore other MDA methods to further improve the WERs
and achieve better parameter efficiency.

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The Inverse Spectral Problem for Jacobi-Type Pencils

The Inverse Spectral Problem for Jacobi-Type Pencils††This paper is a
contribution to the Special Issue on Orthogonal Polynomials, Special Functions
and Applications (OPSFA14). The full collection is available at
https://www.emis.de/journals/SIGMA/OPSFA2017.html

Sergey M. ZAGORODNYUK

S.M. Zagorodnyuk

School of Mathematics and Computer Sciences, V.N. Karazin Kharkiv National
University,
Svobody Square 4, Kharkiv 61022, Ukraine<EMAIL_ADDRESS>
Received June 10, 2017, in final form October 24, 2017; Published online
October 28, 2017

In this paper we study the inverse spectral problem for Jacobi-type pencils.
By a Jacobi-type pencil we mean the following pencil $J_{5}-\lambda J_{3}$,
where $J_{3}$ is a Jacobi matrix and $J_{5}$ is a semi-infinite real symmetric
five-diagonal matrix with positive numbers on the second subdiagonal. In the
case of a special perturbation of orthogonal polynomials on a finite interval
the corresponding spectral function takes an explicit form.

operator pencil; recurrence relation; orthogonal polynomials; spectral
function

42C05; 47B36

##  1 Introduction

The theory of operator pencils in Banach spaces has many applications and it
is actively developing nowadays, see [7, 8, 10]. By operator pencils or
operator polynomials one means polynomials of complex variable $\lambda$ whose
coefficients are linear bounded operators acting in a Banach space $X$

$\displaystyle L(\lambda)=\sum_{j=0}^{m}\lambda^{j}A_{j},$

where $A_{j}\colon X\rightarrow X$, $j=0,\dots,m$. As it was noted in Rodman’s
book [10], there exist three broad topics in this theory which are the main
foci of interest:

  * (a)

linearization, i.e., reduction to a linear polynomial;

  * (b)

various types of factorizations;

  * (c)

the problems of multiple completness of eigenvectors and generalized
eigenvectors.

If $X$ is a Hilbert space and all operators $A_{j}$ are self-adjoint, the
pencil $L(\lambda)$ is said to be self-adjoint. In the case $m=1$ ($m=2$) the
pencil is called linear (respectively quadratic). For the recent progress on
quadratic operator pencils and their applications we refer to the book of
Möller and Pivovarchik [8].

Linear self-adjoint operator pencils has no spectral theory which could be
compatible with the theory of single or commuting self-adjoint operators. The
generalized eigenvalue problem

$\displaystyle(A_{0}+\lambda A_{1})x=0,\qquad x\in X,$

is essentially more generic than the usual eigenvalue problem for a self-
adjoint operator. We can explain this difference by the following astonishing
result from the theory of linear matrix pencils.

######  Theorem 1.1 ([9, Theorem 15.2.1, p. 341]).

Any real square matrix $B$ can be written as $B=AM^{-1}$ or $B=M^{-1}A$ where
$A$ and $M$ are suitable symmetric matrices.

Nevertheless, the general spectral theory for linear operator pencils has old
and new contributions. In particular, it has applications to quotients of
bounded operators and to the study of electron waveguides in graphene, see [5]
and references therein.

Recently, Ben Amara, Vladimirov and Shkalikov investigated the following
linear pencil of differential operators [2]

$\displaystyle(py^{\prime\prime})^{\prime\prime}-\lambda(-y^{\prime\prime}+cry)=0.$
(1.1)

The initial conditions are $y(0)=y^{\prime}(0)=y(1)=y^{\prime}(1)=0$, or
$y(0)=y^{\prime}(0)=y^{\prime}(1)=(py^{\prime\prime})^{\prime}(1)+\lambda\alpha
y(1)=0$. Here $p,r\in C[0,1]$ are uniformly positive, while the parameters $c$
and $\alpha$ are real. Equation (1.1) has several physical applications,
including a motion of a partially fixed bar with additional constraints in the
elasticity theory [2].

It turns out that equation (1.1) is related to certain generalized orthogonal
polynomials, just as the Sturm–Liouville differential operator is related to
the orthogonal polynomials on the real line (OPRL).

The theory of orthogonal polynomials has numerous old and new contributions
and applications, see [4, 6, 11, 12, 13, 14]. This theory is closely related
to spectral problems for Jacobi matrices. The direct and inverse spectral
problems for Jacobi matrices (with matrix elements) were described, e.g., in
[1, 3]. For various other types of spectral problems we refer to historical
notes in [15]. Recall the following basic definition from [16].

######  Definition 1.2.

A set $\Theta=(J_{3},J_{5},\alpha,\beta)$, where $\alpha>0$,
$\beta\in\mathbb{R}$, $J_{3}$ is a Jacobi matrix and $J_{5}$ is a semi-
infinite real symmetric five-diagonal matrix with positive numbers on the
second subdiagonal, is said to be a Jacobi-type pencil $($of matrices$)$.

From this definition we see that matrices $J_{3}$ and $J_{5}$ have the
following form

$\displaystyle J_{3}=\left(\begin{matrix}b_{0}&a_{0}&0&0&0&\cdots\\\
a_{0}&b_{1}&a_{1}&0&0&\cdots\\\ 0&a_{1}&b_{2}&a_{2}&0&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{matrix}\right),\qquad a_{k}>0,\quad
b_{k}\in\mathbb{R},\quad k\in\mathbb{Z}_{+},$ (1.2) $\displaystyle
J_{5}=\left(\begin{matrix}\alpha_{0}&\beta_{0}&\gamma_{0}&0&0&0&\cdots\\\
\beta_{0}&\alpha_{1}&\beta_{1}&\gamma_{1}&0&0&\cdots\\\
\gamma_{0}&\beta_{1}&\alpha_{2}&\beta_{2}&\gamma_{2}&0&\cdots\\\
0&\gamma_{1}&\beta_{2}&\alpha_{3}&\beta_{3}&\gamma_{3}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{matrix}\right),\qquad\alpha_{n},\beta_{n}\in\mathbb{R},\quad\gamma_{n}>0,\quad
n\in\mathbb{Z}_{+}.$ (1.3)

With a Jacobi-type pencil of matrices $\Theta$ one associates a system of
polynomials $\\{p_{n}(\lambda)\\}_{n=0}^{\infty}$, such that

$\displaystyle p_{0}(\lambda)=1,\qquad p_{1}(\lambda)=\alpha\lambda+\beta,$

and

$\displaystyle(J_{5}-\lambda J_{3})\vec{p}(\lambda)=0,$ (1.4)

where
$\vec{p}(\lambda)=(p_{0}(\lambda),p_{1}(\lambda),p_{2}(\lambda),\dots)^{\rm
T}$. Here the superscript ${\rm T}$ means the transposition. Polynomials
$\\{p_{n}(\lambda)\\}_{n=0}^{\infty}$ are said to be associated to the Jacobi-
type pencil of matrices $\Theta$.

Observe that for each system of orthonormal polynomials on the real line with
$p_{0}=1$ one may choose $J_{3}$ to be the corresponding Jacobi matrix (which
elements are the recurrence coefficients), $J_{5}=J_{3}^{2}$, and $\alpha$,
$\beta$ being the coefficients of $p_{1}$
($p_{1}(\lambda)=\alpha\lambda+\beta$).

One can rewrite relation (1.4) in the scalar form

$\displaystyle\gamma_{n-2}p_{n-2}(\lambda)+(\beta_{n-1}-\lambda
a_{n-1})p_{n-1}(\lambda)+(\alpha_{n}-\lambda b_{n})p_{n}(\lambda)$
$\displaystyle\qquad{}+(\beta_{n}-\lambda
a_{n})p_{n+1}(\lambda)+\gamma_{n}p_{n+2}(\lambda)=0,\qquad
n\in\mathbb{Z}_{+},$ (1.5)

where $p_{-2}(\lambda)=p_{-1}(\lambda)=0$,
$\gamma_{-2}=\gamma_{-1}=a_{-1}=\beta_{-1}=0$.

Let us return to the differential equation (1.1) and explain how it is related
to polynomials from (1.5). The idea is the same as in the classical case. We
consider a partition of the interval $[0,1]$

$\displaystyle 0=x_{0}<x_{1}<x_{2}<\cdots<x_{N-1}<x_{N}=1,$

with a uniform step $h$. Set $y_{j}:=y(x_{j})$, $p_{j}:=p(x_{j})$,
$r_{j}:=r(x_{j})$, $0\leq j\leq N$. Then we replace the derivatives in (1.1)
by the corresponding discretizations
$y^{\prime}(x_{j})\approx\frac{y_{j+1}-y_{j}}{h}$,
$y^{\prime\prime}(x_{j})\approx\frac{y_{j+1}-2y_{j}+y_{j-1}}{h^{2}}$. We
obtain the following equation

$\displaystyle
p_{j-1}y_{j-2}-2(p_{j}+p_{j-1})y_{j-1}+(p_{j+1})+4p_{j}+p_{j-1})y_{j}-2(p_{j+1}+p_{j})y_{j+1}$
$\displaystyle\qquad{}+p_{j+1}y_{j+2}-\lambda
h^{2}\big{(}y_{j-1}-\big{(}2-h^{2}cr_{j}\big{)}y_{j}+y_{j+1}\big{)}=0.$

Denote

$\displaystyle\alpha_{j}=p_{j+1}+4p_{j}+p_{j-1},\qquad\beta_{j}=-2(p_{j+1}+p_{j}),\qquad\gamma_{j}=p_{j+1},$
$\displaystyle a_{j}=h^{2},\qquad b_{j}=\big{(}{-}2-h^{2}cr_{j}\big{)}h^{2}.$

Then we get

$\displaystyle\gamma_{j-2}y_{j-2}+\beta_{j-1}y_{j-1}+\alpha_{j}y_{j}+\beta_{j}y_{j+1}+\gamma_{j}y_{j+2}+\lambda\big{(}a_{j-1}y_{j-1}+b_{j}y_{j}+a_{j}y_{j+1}\big{)}=0.$

If we set $\widetilde{\lambda}=-\lambda$, we obtain a relation of the form
(1.5). Thus, relation (1.5) forms a discrete grid model for the equation
(1.1). The boundary conditions $y(0)=0$, $y^{\prime}(0)=0$ correspond to the
convention $y_{-2}=y_{-1}=0$. Instead of $[0,1]$ we can consider
$[0,+\infty)$. The conditions on the right end of the bar we replace by
conditions on the left end which are controlled by $\alpha$ and $\beta$.

In this paper we shall introduce and study the direct and inverse spectral
problems for Jacobi type pencils. We shall also consider a special
perturbation of the class of orthonormal polynomials on a finite real
interval. In that case it is possible to obtain a more convenient integral
representation for the corresponding spectral function.

Notations. As usual, we denote by $\mathbb{R}$, $\mathbb{C}$, $\mathbb{N}$,
$\mathbb{Z}$, $\mathbb{Z}_{+}$, the sets of real numbers, complex numbers,
positive integers, integers and non-negative integers, respectively. By
$\mathbb{P}$ we denote the set of all polynomials with complex coefficients.

By $l_{2}$ we denote the usual Hilbert space of all complex sequences
$c=(c_{n})_{n=0}^{\infty}=(c_{0},c_{1},c_{2},\dots)^{\rm T}$ with the finite
norm $\|c\|_{l_{2}}=\sqrt{\sum\limits_{n=0}^{\infty}|c_{n}|^{2}}$. Here $T$
means the transposition. The scalar product of two sequences
$c=(c_{n})_{n=0}^{\infty}$, $d=(d_{n})_{n=0}^{\infty}\in l_{2}$ is given by
$(c,d)_{l_{2}}=\sum\limits_{n=0}^{\infty}c_{n}\overline{d_{n}}$. We denote
$\vec{e}_{m}=(\delta_{n,m})_{n=0}^{\infty}\in l_{2}$, $m\in\mathbb{Z}_{+}$. By
$l_{2,{\rm f\/in}}$ we denote the set of all finite vectors from $l_{2}$,
i.e., vectors with all, but finite number, components being zeros. By
$\operatorname{diag}(c_{1},c_{2},c_{3},\dots)$ we mean a semi-infinite matrix
with $c_{j}\in\mathbb{C}$ in $j$-th column and $j$-th row and zeros outside
the main diagonal.

By $\mathfrak{B}(\mathbb{R})$ we denote the set of all Borel subsets of
$\mathbb{R}$. If $\sigma$ is a (non-negative) bounded measure on
$\mathfrak{B}(\mathbb{R})$ then by $L^{2}_{\sigma}$ we denote a Hilbert space
of all (classes of equivalences of) complex-valued functions $f$ on
$\mathbb{R}$ with a finite norm
$\|f\|_{L^{2}_{\sigma}}=\sqrt{\int_{\mathbb{R}}|f(x)|^{2}{\rm d}\sigma}$. The
scalar product of two functions $f,g\in L^{2}_{\sigma}$ is given by
$(f,g)_{L^{2}_{\sigma}}=\int_{\mathbb{R}}f(x)\overline{g(x)}{\rm d}\sigma$. By
$[f]$ we denote the class of equivalence in $L^{2}_{\sigma}$ which contains
the representative $f$. By $\mathcal{P}$ we denote a set of all (classes of
equivalence which contain) polynomials in $L^{2}_{\sigma}$. As usual, we
sometimes use the representatives instead of their classes in formulas. Let
$B$ be an arbitrary linear operator in $L^{2}_{\sigma}$ with the domain
$\mathcal{P}$. Let $f(\lambda)\in\mathbb{P}$ be nonzero and of degree
$d\in\mathbb{Z}_{+}$, $f(\lambda)=\sum\limits_{k=0}^{d}d_{k}\lambda^{k}$,
$d_{k}\in\mathbb{C}$. We set

$\displaystyle f(B)=\sum_{k=0}^{d}d_{k}B^{k},\qquad B^{0}:=E|_{\mathcal{P}}.$

If $f\equiv 0$, then $f(B):=0|_{\mathcal{P}}$.

If H is a Hilbert space then $(\cdot,\cdot)_{H}$ and $\|\cdot\|_{H}$ mean the
scalar product and the norm in $H$, respectively. Indices may be omitted in
obvious cases. For a linear operator $A$ in $H$, we denote by $D(A)$ its
domain, by $R(A)$ its range, by $\operatorname{Ker}A$ its null subspace
(kernel), and $A^{*}$ means the adjoint operator if it exists. If $A$ is
invertible then $A^{-1}$ means its inverse. $\overline{A}$ means the closure
of the operator, if the operator is closable. If $A$ is bounded then $\|A\|$
denotes its norm. For a set $M\subseteq H$ we denote by $\overline{M}$ the
closure of $M$ in the norm of $H$. By $\operatorname{Lin}M$ we mean the set of
all linear combinations of elements of $M$, and
$\overline{\operatorname{span}}\,M:=\overline{\operatorname{Lin}M}$. By
$E=E_{H}$ ($0=0_{H}$) we denote the identity operator in $H$, i.e.,
$E_{H}x=x$, $x\in H$ (respectively the null operator in $H$, i.e., $0_{H}x=0$,
$x\in H$). If $H_{1}$ is a subspace of $H$, then $P_{H_{1}}=P_{H_{1}}^{H}$ is
an operator of the orthogonal projection on $H_{1}$ in $H$.

##  2 Preliminaries

In this section, for the convenience of the reader, we recall basic
definitions and results from [16]. Then we state the direct and inverse
spectral problems for the Jacobi-type pencils. The direct problem will be
solved immediately, while the inverse problem will be considered in the next
section. Set

$\displaystyle
u_{n}:=J_{3}\vec{e}_{n}=a_{n-1}\vec{e}_{n-1}+b_{n}\vec{e}_{n}+a_{n}\vec{e}_{n+1},$
$\displaystyle
w_{n}:=J_{5}\vec{e}_{n}=\gamma_{n-2}\vec{e}_{n-2}+\beta_{n-1}\vec{e}_{n-1}+\alpha_{n}\vec{e}_{n}+\beta_{n}\vec{e}_{n+1}+\gamma_{n}\vec{e}_{n+2},\qquad
n\in\mathbb{Z}_{+}.$

Here and in what follows by $\vec{e}_{k}$ with negative $k$ we mean (vector)
zero. The following operator

$\displaystyle
Af=\frac{\zeta}{\alpha}(\vec{e}_{1}-\beta\vec{e}_{0})+\sum_{n=0}^{\infty}\xi_{n}w_{n},$
$\displaystyle f=\zeta\vec{e}_{0}+\sum_{n=0}^{\infty}\xi_{n}u_{n}\in l_{2,{\rm
f\/in}},\qquad\zeta,\xi_{n}\in\mathbb{C},$ (2.1)

with $D(A)=l_{2,{\rm f\/in}}$ is said to be the associated operator for the
Jacobi-type pencil $\Theta$. Notice that in the sums in (2.1) only finite
number of $\xi_{n}$ are nonzero. We shall always assume this in the case of
elements from the linear span. In particular, we have

$\displaystyle AJ_{3}\vec{e}_{n}=J_{5}\vec{e}_{n},\qquad n\in\mathbb{Z}_{+},$

and therefore

$\displaystyle AJ_{3}=J_{5}.$

As usual, the matrices $J_{3}$ and $J_{5}$ define linear operators with the
domain $l_{2,{\rm f\/in}}$ which we denote by the same letters.

For an arbitrary non-zero polynomial $f(\lambda)\in\mathbb{P}$ of degree
$d\in\mathbb{Z}_{+}$, $f(\lambda)=\sum\limits_{k=0}^{d}d_{k}\lambda^{k}$,
$d_{k}\in\mathbb{C}$, we set $f(A)=\sum\limits_{k=0}^{d}d_{k}A^{k}$. Here
$A^{0}:=E|_{l_{2,{\rm f\/in}}}$. For $f(\lambda)\equiv 0$, we set
$f(A)=0|_{l_{2,{\rm f\/in}}}$. The following relations hold [16]

$\displaystyle\vec{e}_{n}=p_{n}(A)\vec{e}_{0},\qquad n\in\mathbb{Z}_{+},$
(2.2)
$\displaystyle\big{(}p_{n}(A)\vec{e}_{0},p_{m}(A)\vec{e}_{0}\big{)}_{l_{2}}=\delta_{n,m},\qquad
n,m\in\mathbb{Z}_{+}.$ (2.3)

Denote by $\\{r_{n}(\lambda)\\}_{n=0}^{\infty}$, $r_{0}(\lambda)=1$, the
system of polynomials satisfying

$\displaystyle
J_{3}\vec{r}(\lambda)=\lambda\vec{r}(\lambda),\qquad\vec{r}(\lambda)=(r_{0}(\lambda),r_{1}(\lambda),r_{2}(\lambda),\dots)^{\rm
T}.$ (2.4)

These polynomials are orthonormal on the real line with respect to a (possibly
non-unique) non-negative finite measure $\sigma$ on $\mathfrak{B}(\mathbb{R})$
(Favard’s theorem). Consider the following operator

$\displaystyle
U\sum_{n=0}^{\infty}\xi_{n}\vec{e}_{n}=\left[\sum_{n=0}^{\infty}\xi_{n}r_{n}(x)\right],\qquad\xi_{n}\in\mathbb{C},$
(2.5)

which maps $l_{2,{\rm f\/in}}$ onto $\mathcal{P}$. Here by $\mathcal{P}$ we
denote a set of all (classes of equivalence which contain) polynomials in
$L^{2}_{\sigma}$. Denote

$\displaystyle\mathcal{A}=\mathcal{A}_{\sigma}=UAU^{-1}.$ (2.6)

The operator $\mathcal{A}=\mathcal{A}_{\sigma}$ is said to be the model
representation in $L^{2}_{\sigma}$ of the associated operator $A$.

######  Theorem 2.1 ([16]).

Let $\Theta=(J_{3},J_{5},\alpha,\beta)$ be a Jacobi-type pencil. Let
$\\{r_{n}(\lambda)\\}_{n=0}^{\infty}$, $r_{0}(\lambda)=1$, be a system of
polynomials satisfying (2.4) and $\sigma$ be their $($arbitrary$)$
orthogonality measure on $\mathfrak{B}(\mathbb{R})$. The associated
polynomials $\\{p_{n}(\lambda)\\}_{n=0}^{\infty}$ satisfy the following
relations

$\displaystyle\int_{\mathbb{R}}p_{n}(\mathcal{A})(1)\overline{p_{m}(\mathcal{A})(1)}{\rm
d}\sigma=\delta_{n,m},\qquad n,m\in\mathbb{Z}_{+},$ (2.7)

where $\mathcal{A}$ is the model representation in $L^{2}_{\sigma}$ of the
associated operator $A$.

We have now recalled all basic notations and results which we shall need in
the present paper. We introduce the following definition.

######  Definition 2.2.

Let $\Theta=(J_{3},J_{5},\alpha,\beta)$ be a Jacobi-type pencil and
$\\{p_{n}(\lambda)\\}_{n=0}^{\infty}$ be the associated polynomials to
$\Theta$. A sesquilinear functional $S(u,v)$, $u,v\in\mathbb{P}$, satisfying
the following relation

$\displaystyle S(p_{n},p_{m})=\delta_{n,m},\qquad n,m\in\mathbb{Z}_{+},$

is said to be the spectral function of the Jacobi-type pencil $\Theta$.

In a usual manner, we may introduce the corresponding direct and inverse
spectral problems. The direct spectral problem for a Jacobi-type pencil
$\Theta$ consists in searching for answers on the following questions:

  * 1)

Does the spectral function exist?

  * 2)

If the spectral function exists, is it unique?

  * 3)

If the spectral function exists, how to find it (or them)?

By relation (2.3) we see that the spectral function always exists. By the
linearity it follows that the spectral function is unique. It has the
following representation

$\displaystyle
S(u,v)=\big{(}u(A)\vec{e}_{0},v(A)\vec{e}_{0}\big{)}_{l_{2}},\qquad
u,v\in\mathbb{P},$ (2.8)

where $A$ is the associated operator for $\Theta$. By (2.8) we obtain that

$\displaystyle\overline{S(u,v)}=S(v,u),\qquad u,v\in\mathbb{P},$ (2.9)

and

$\displaystyle S(u,u)\geq 0,\qquad u\in\mathbb{P}.$ (2.10)

Moreover, the following integral representation holds

$\displaystyle
S(u,v)=\int_{\mathbb{R}}u(\mathcal{A}_{\sigma})(1)\overline{v(\mathcal{A}_{\sigma})(1)}{\rm
d}\sigma,\qquad u,v\in\mathbb{P}.$ (2.11)

Thus, the direct spectral problem for $\Theta$ is solved in full.

The inverse spectral problem for a Jacobi-type pencil $\Theta$ consists in
searching for answers on the following questions:

  * (a)

Is it possible to reconstruct the Jacobi-type pencil
$\Theta=(J_{3},J_{5},\alpha,\beta)$ using its spectral function? If it is
possible, what is the procedure of the reconstruction?

  * (b)

What are necessary and sufficient conditions for a sesquilinear functional
$\sigma(u,v)$, $u,v\in\mathbb{P}$, to be the spectral function of a Jacobi-
type pencil $\Theta=(J_{3},J_{5},\alpha,\beta)$?

Questions (a) and (b) will be investigated in the next section.

##  3 The inverse spectral problem for a Jacobi-type pencil

An answer to the question (b) of the inverse spectral problem is given by the
following theorem.

######  Theorem 3.1.

A sesquilinear functional $S(u,v)$, $u,v\in\mathbb{P}$, satisfying relations
(2.9), (2.10), is the spectral function of a Jacobi-type pencil if and only if
it admits the following integral representation

$\displaystyle
S(u,v)=\int_{\mathbb{R}}u(\mathcal{A})(1)\overline{v(\mathcal{A})(1)}{\rm
d}\sigma,\qquad u,v\in\mathbb{P},$ (3.1)

where $\sigma$ is a non-negative measure on $\mathfrak{B}(\mathbb{R})$ with
all finite power moments,

$\displaystyle\int_{\mathbb{R}}{\rm
d}\sigma=1,\qquad\int_{\mathbb{R}}|g(x)|^{2}{\rm d}\sigma>0,$

for any non-zero complex polynomial $g$, and $\mathcal{A}$ is a linear
operator in $L^{2}_{\sigma}$ with the following properties:

  * $(i)$

$D(\mathcal{A})=\mathcal{P}$;

  * $(ii)$

for each $k\in\mathbb{Z}_{+}$ it holds

$\displaystyle\mathcal{A}x^{k}=\xi_{k,k+1}x^{k+1}+\sum_{j=0}^{k}\xi_{k,j}x^{j},$
(3.2)

where $\xi_{k,k+1}>0$, $\xi_{k,j}\in\mathbb{R}$, $0\leq j\leq k$;

  * $(iii)$

the operator $\mathcal{A}\Lambda_{0}$ is symmetric, here by $\Lambda_{0}$ we
denote the operator of the multiplication by an independent variable in
$L^{2}_{\sigma}$ restricted to $\mathcal{P}$.

###### Proof.

Necessity. Let $\Theta=(J_{3},J_{5},\alpha,\beta)$ be a Jacobi-type pencil and
$A$ be the associated operator of $\Theta$. Define polynomials
$\\{r_{n}(\lambda)\\}$, the measure $\sigma$, the operators $U$ and
$\mathcal{A}=\mathcal{A}_{\sigma}$ as in the introduction, see (2.4), (2.5),
(2.6). Let $S(u,v)$ be the spectral function of the pencil $\Theta$. It has an
integral representation (2.11). It remains to check that $\sigma$ and
$\mathcal{A}$ possess all the required properties in the statement of the
theorem.

Since $\sigma$ is generated by the Jacobi matrix $J_{3}$, it has all required
properties as in the statement of the theorem. The operator
$\mathcal{A}_{\sigma}$ is linear and defined on $\mathcal{P}$.

######  Lemma 3.2.

The associated operator $A$ of a Jacobi-type pencil $\Theta$ has the following
properties:

  * $1)$

for each $n\in\mathbb{Z}_{+}$ it holds

$\displaystyle\vec{e}_{n}=d_{n,n}A^{n}\vec{e}_{0}+\sum_{j=0}^{n-1}d_{n,j}A^{j}\vec{e}_{0},$
(3.3)

where $d_{n,n}>0$, $d_{n,j}\in\mathbb{R}$, $0\leq j\leq n-1$;

  * $2)$

for each $n\in\mathbb{Z}_{+}$ it holds

$\displaystyle
A^{n}\vec{e}_{0}=c_{n,n}\vec{e}_{n}+\sum_{j=0}^{n-1}c_{n,j}\vec{e}_{j},$ (3.4)

where $c_{n,n}>0$, $c_{n,j}\in\mathbb{R}$, $0\leq j\leq n-1$;

  * $3)$

for each $n\in\mathbb{Z}_{+}$ it holds

$\displaystyle
A\vec{e}_{n}=f_{n,n+1}\vec{e}_{n+1}+\sum_{j=0}^{n}f_{n,j}\vec{e}_{j},$ (3.5)

where $f_{n,n+1}>0$, $f_{n,j}\in\mathbb{R}$, $0\leq j\leq n$.

###### Proof of Lemma 3.2.

Let $p_{n}(\lambda)=\sum\limits_{j=0}^{n}d_{n,j}\lambda^{j}$, $d_{n,n}>0$,
$d_{n,j}\in\mathbb{R}$, be the associated polynomials of the pencil. By (2.2)
we conclude that relation (3.3) holds.

Let us check relation (3.4) by the induction argument. For the cases $n=0,1$
relation (3.4) obviously holds, see the definition of $A$. Assume that (3.4)
holds for $n=1,2,\dots,r$, $r\in\mathbb{N}$. By (3.3) with $n=r+1$ we may
write

$\displaystyle
A^{r+1}\vec{e}_{0}=\frac{1}{d_{r+1,r+1}}\vec{e}_{r+1}-\frac{1}{d_{r+1,r+1}}\sum_{j=0}^{r}d_{r+1,j}A^{j}\vec{e}_{0}$
$\displaystyle\hphantom{A^{r+1}\vec{e}_{0}}{}=\frac{1}{d_{r+1,r+1}}\vec{e}_{r+1}+\sum_{j=0}^{r}\sum_{m=0}^{j}\frac{(-1)d_{r+1,j}c_{j,m}}{d_{r+1,r+1}}\vec{e}_{m}.$

Finally, in order to prove relation (3.5) we apply the operator $A$ to the
both sides of relation (3.3) and use (3.4)

$\displaystyle
A\vec{e}_{n}=d_{n,n}A^{n+1}\vec{e}_{0}+\sum_{j=0}^{n-1}d_{n,j}A^{j+1}\vec{e}_{0}$
$\displaystyle\hphantom{A\vec{e}_{n}}{}=d_{n,n}c_{n+1,n+1}\vec{e}_{n+1}+\sum_{j=0}^{n}d_{n,n}c_{n+1,j}\vec{e}_{j}+\sum_{j=0}^{n-1}\sum_{k=0}^{j+1}d_{n,j}c_{j+1,k}\vec{e}_{k},\qquad
n\in\mathbb{Z}_{+}.$

Lemma 3.2 is proved. ∎

Return to the proof of Theorem 3.1. Choose an arbitrary $k\in\mathbb{Z}_{+}$.
Observe that

$\displaystyle x^{k}=\sum_{j=0}^{k}g_{k,j}r_{j}(x),\qquad
g_{k,j}\in\mathbb{R},\quad g_{k,k}>0.$

By (3.5) we may write

$\displaystyle\mathcal{A}_{\sigma}\big{[}x^{k}\big{]}=\sum_{j=0}^{k}g_{k,j}UA\vec{e}_{j}=\sum_{j=0}^{k}g_{k,j}\left(f_{j,j+1}[r_{j+1}]+\sum_{l=0}^{j}f_{j,l}[r_{l}]\right)$
$\displaystyle\hphantom{\mathcal{A}_{\sigma}\big{[}x^{k}\big{]}}{}=g_{k,k}f_{k,k+1}[r_{k+1}]+\left[\sum_{j=0}^{k-1}g_{k,j}f_{j,j+1}r_{j+1}+\sum_{j=0}^{k}g_{k,j}\sum_{l=0}^{j}f_{j,l}r_{l}\right]$
$\displaystyle\hphantom{\mathcal{A}_{\sigma}\big{[}x^{k}\big{]}}{}=\left[g_{k,k}f_{k,k+1}\widetilde{\mu_{k+1,k+1}}x^{k+1}+g_{k,k}f_{k,k+1}\sum_{m=0}^{k}\widetilde{\mu_{k+1,m}}x^{m}\right.$
$\displaystyle\left.\hphantom{\mathcal{A}_{\sigma}\big{[}x^{k}\big{]}=}{}+\sum_{j=0}^{k-1}g_{k,j}f_{j,j+1}r_{j+1}+\sum_{j=0}^{k}g_{k,j}\sum_{l=0}^{j}f_{j,l}r_{l}\right],$
(3.6)

where we set
$r_{l}(\lambda)=\sum\limits_{j=0}^{l}\widetilde{\mu}_{l,j}\lambda^{j}$,
$\widetilde{\mu}_{l,l}>0$, $\mu_{l,j}\in\mathbb{R}$. Since we get a real
polynomial of degree $k+1$ on the right-hand side of (3.6) and it has a
positive leading coefficient $g_{k,k}f_{k,k+1}\widetilde{\mu_{k+1,k+1}}$, then
relation (3.2) is proved.

It remains to verify $(iii)$. For arbitrary $n,m\in\mathbb{Z}_{+}$ we may
write

$\displaystyle(\mathcal{A}_{\sigma}\Lambda_{0}[r_{n}(x)],[r_{m}(x)])=\big{(}UAU^{-1}[a_{n-1}r_{n-1}(x)+b_{n}r_{n}(x)+a_{n}r_{n+1}(x)],[r_{m}(x)]\big{)}$
$\displaystyle\hphantom{(\mathcal{A}_{\sigma}\Lambda_{0}[r_{n}(x)],[r_{m}(x)])}{}=(AJ_{3}\vec{e}_{n},\vec{e}_{m})=(J_{5}\vec{e}_{n},\vec{e}_{m})=(\vec{e}_{n},J_{5}\vec{e}_{m})$
$\displaystyle\hphantom{(\mathcal{A}_{\sigma}\Lambda_{0}[r_{n}(x)],[r_{m}(x)])}{}=(\vec{e}_{n},A(a_{m-1}\vec{e}_{m-1}+b_{m}\vec{e}_{m}+a_{m}\vec{e}_{m+1}))$
$\displaystyle\hphantom{(\mathcal{A}_{\sigma}\Lambda_{0}[r_{n}(x)],[r_{m}(x)])}{}=([r_{n}(x)],\mathcal{A}_{\sigma}\Lambda_{0}[r_{m}(x)]).$
(3.7)

By the linearity we conclude that $\mathcal{A}_{\sigma}\Lambda_{0}$ is
symmetric.

Sufficiency. Suppose that a sesquilinear functional $S(u,v)$,
$u,v\in\mathbb{P}$, satisfying relations (2.9), (2.10), is given. Assume that
it has the integral representation (3.1) where $\sigma$ is a non-negative
probability measure on $\mathfrak{B}(\mathbb{R})$ with all finite moments,
$\int_{\mathbb{R}}|g(x)|^{2}{\rm d}\sigma>0$, for any non-zero complex
polynomial $g$, and $\mathcal{A}$ is a linear operator in $L^{2}_{\sigma}$
with properties $(i)$–$(iii)$. By condition $(ii)$ and the induction argument
it can be directly verified that for each $n\in\mathbb{Z}_{+}$ it holds

$\displaystyle\mathcal{A}^{n}[1]=a_{n,n}[x^{n}]+\sum_{j=0}^{n-1}a_{n,j}\big{[}x^{j}\big{]},$

where $a_{n,n}>0$, $a_{n,j}\in\mathbb{R}$, $0\leq j\leq n-1$.

Suppose that $S(u,u)=0$ for a complex polynomial $u$. Then

$\displaystyle 0=\|u(\mathcal{A})(1)\|^{2}_{L^{2}_{\sigma}}.$

Assume that $u$ is nonzero,
$u(\lambda)=\sum\limits_{k=0}^{n}d_{k}\lambda^{k}$, $d_{n}\not=0$,
$d_{k}\in\mathbb{C}$, $n\in\mathbb{Z}_{+}$. Observe that

$\displaystyle
u(\mathcal{A})(1)=\left(\sum_{k=0}^{n}d_{k}\mathcal{A}^{k}\right)[1]=d_{n}\mathcal{A}^{n}[1]+\sum_{k=0}^{n-1}d_{k}\mathcal{A}^{k}[1]$
$\displaystyle\hphantom{u(\mathcal{A})(1)}{}=d_{n}\left(a_{n,n}\big{[}x^{n}\big{]}+\left[\sum_{j=0}^{n-1}a_{n,j}x^{j}\right]\right)+\sum_{k=0}^{n-1}d_{k}\sum_{j=0}^{k}a_{k,j}\big{[}x^{j}\big{]}$
$\displaystyle\hphantom{u(\mathcal{A})(1)}{}=\left[d_{n}a_{n,n}x^{n}+d_{n}\sum_{j=0}^{n-1}a_{n,j}x^{j}+\sum_{k=0}^{n-1}\sum_{j=0}^{k}d_{k}a_{k,j}x^{j}\right]=:[r(x)].$

We have a nonzero polynomial $r$ (of degree $n$) with
$\|r\|_{L^{2}_{\sigma}}=0$. This contradicts to our assumptions on the measure
$\sigma$. Consequently, $u\equiv 0$.

The functional $S$ defines an inner product on $\mathbb{P}$. Thus, the complex
vector space $\mathbb{P}$ becomes a space $H$ with a scalar product. It is a
normed space with the norm $\|p\|=\sqrt{S(p,p)}$. We shall not need its
completion. Set

$\displaystyle\mathbf{p}_{0}(\lambda)=\frac{1}{\|1\|_{H}}=1,\qquad\mathbf{p}_{1}(\lambda)=\frac{\lambda-S(x,1)1}{\|\lambda-S(x,1)1\|_{H}}.$

Let us check that $\mathbf{p}_{1}(\lambda)$ is well-defined. Denote by
$\\{s_{k}\\}_{k=0}^{\infty}$ the power moments of $\sigma$

$\displaystyle s_{k}=\int_{\mathbb{R}}x^{k}{\rm d}\sigma,\qquad
k\in\mathbb{Z}_{+}.$

Let

$\displaystyle\Delta_{n}:=\det(s_{k+l})_{k,l=0}^{n},\qquad
n\in\mathbb{Z}_{+},\qquad\Delta_{-1}:=1,$

be the corresponding Hankel determinants. Observe that

$\displaystyle S(x,1)=\int\mathcal{A}(1){\rm
d}\sigma=\int(\xi_{0,1}\lambda+\xi_{0,0}){\rm
d}\sigma=\xi_{0,1}s_{1}+\xi_{0,0}\in\mathbb{R},$ $\displaystyle
S(x,x)=\int\mathcal{A}(1)\overline{\mathcal{A}(1)}{\rm
d}\sigma=\int(\xi_{0,1}\lambda+\xi_{0,0})^{2}{\rm
d}\sigma=\xi_{0,1}^{2}s_{2}+2\xi_{0,1}\xi_{0,0}s_{1}+\xi_{0,0}^{2}.$

We may write

$\displaystyle\|\lambda-S(x,1)1\|_{H}^{2}=S(\lambda-S(x,1)1,\lambda-S(x,1)1)=S(\lambda,\lambda)-(S(\lambda,1))^{2}$
$\displaystyle\hphantom{\|\lambda-S(x,1)1\|_{H}^{2}}{}=\xi_{0,1}^{2}\big{(}s_{2}-s_{1}^{2}\big{)}=\xi_{0,1}^{2}\Delta_{1}>0,$

where we have used our assumptions on the measure $\sigma$. Then

$\displaystyle\mathbf{p}_{1}(\lambda)=\frac{\lambda-\xi_{0,1}s_{1}-\xi_{0,0}}{\xi_{0,1}\sqrt{\Delta_{1}}}.$

Set

$\displaystyle\alpha=\frac{1}{\xi_{0,1}\sqrt{\Delta_{1}}},\qquad\beta=-\frac{\xi_{0,1}s_{1}+\xi_{0,0}}{\xi_{0,1}\sqrt{\Delta_{1}}}.$
(3.8)

Let $\\{\mathbf{r}_{n}(\lambda)\\}_{n=0}^{\infty}$ be orthonormal polynomials
with respect to the measure $\sigma$ (having positive leading coefficients).
Denote by $J_{3}$ the corresponding Jacobi matrix (formed by the recurrence
coefficients of $\mathbf{r}_{n}$). Denote

$\displaystyle J_{5}=(g_{m,n})_{m,n=0}^{\infty},\qquad
g_{m,n}:=(\mathcal{A}\Lambda_{0}[\mathbf{r}_{n}(\lambda)],[\mathbf{r}_{m}(\lambda)])_{L^{2}_{\sigma}}.$

By condition $(iii)$ of the theorem we conclude that $J_{5}$ is a symmetric
semi-infinite matrix. Let

$\displaystyle\mathbf{r}_{n}(\lambda)=\sum_{k=0}^{n}\eta_{n,k}\lambda^{k},\qquad\eta_{n,k}\in\mathbb{R},\quad\eta_{n,n}>0,\quad
n\in\mathbb{Z}_{+}.$

For an arbitrary $n\in\mathbb{Z}_{+}$ by condition $(ii)$ we may write

$\displaystyle\mathcal{A}\Lambda_{0}[\mathbf{r}_{n}(\lambda)]=\sum_{k=0}^{n}\eta_{n,k}\mathcal{A}\big{[}\lambda^{k+1}\big{]}=\big{[}\eta_{n,n}\xi_{n+1,n+2}\lambda^{n+2}+d_{n+1}(\lambda)\big{]},$
(3.9)

where $d_{n+1}(\lambda)$ is a zero polynomial or a polynomial with real
coefficients, $\deg p\leq n+1$. Then

$\displaystyle g_{m,n}=0,\qquad m,n\in\mathbb{Z}_{+},\quad m>n+2,$

and

$\displaystyle
g_{n+2,n}=\eta_{n,n}\xi_{n+1,n+2}\big{(}\big{[}\lambda^{n+2}\big{]},[r_{n+2}(\lambda)]\big{)}=\frac{\eta_{n,n}\xi_{n+1,n+2}}{\eta_{n+2,n+2}}>0,\qquad
n\in\mathbb{Z}_{+}.$

By (3.9) we also see that $g_{m,n}$ are real numbers. We conclude that $J_{5}$
is real five-diagonal and it has positive numbers on the second sub-diagonal.

Consider a Jacobi-type pencil $\widetilde{\Theta}=(J_{3},J_{5},\alpha,\beta)$.
Let $\\{p_{n}(\lambda)\\}_{n=0}^{\infty}$ be the associated polynomials to
$\widetilde{\Theta}$. For the pencil $\widetilde{\Theta}$ we define the
standard objects from the introduction. The polynomials $r_{n}(\lambda)$ (see
(2.4)) coincide with $\mathbf{r}_{n}(\lambda)$. We may also choose the given
$\sigma$ as the orthogonality measure (the orthogonality measure can be non-
unique). The operators $A$, $U$ and $\mathcal{A}_{\sigma}$ we define in the
standard way, see (2.1), (2.5), (2.6). We only can not use the brief notation
$\mathcal{A}$ for $\mathcal{A}_{\sigma}$, since $\mathcal{A}$ already denotes
the operator in the integral representation of $S$.

For the pencil $\widetilde{\Theta}$ we may apply our arguments in the proved
Necessity. By relation (3.7) we may write

$\displaystyle(\mathcal{A}_{\sigma}\Lambda_{0}[r_{n}],[r_{m}])_{L^{2}_{\sigma}}=(J_{5}\vec{e}_{n},\vec{e}_{m})_{l_{2}}=g_{m,n}=(\mathcal{A}\Lambda_{0}[r_{n}],[r_{m}])_{L^{2}_{\sigma}},\qquad
n,m\in\mathbb{Z}_{+}.$

Here the last equality follows by the definition of the matrix $J_{5}$.
Therefore

$\displaystyle\mathcal{A}_{\sigma}[\lambda p(\lambda)]=\mathcal{A}[\lambda
p(\lambda)],\qquad\forall\,p\in\mathbb{P}.$ (3.10)

Notice that

$\displaystyle\mathcal{A}_{\sigma}[1]=UA\vec{e}_{0}=\frac{1}{\alpha}U(\vec{e}_{1}-\beta\vec{e}_{0})=\left[\frac{1}{\alpha}(r_{1}(\lambda)-\beta)\right].$
(3.11)

The orthonormal polynomial $r_{1}(\lambda)$ has the following form

$\displaystyle r_{1}(\lambda)=\frac{1}{\sqrt{\Delta_{1}}}(\lambda-s_{1}).$
(3.12)

By (3.11), (3.12) and (3.8) we conclude that

$\displaystyle\mathcal{A}_{\sigma}[1]=[\xi_{0,1}\lambda+\xi_{0,0}].$

On the other hand, by property $(ii)$ we have
$\mathcal{A}[1]=[\xi_{0,1}\lambda+\xi_{0,0}]$. Therefore

$\displaystyle\mathcal{A}_{\sigma}[1]=\mathcal{A}[1].$ (3.13)

Relations (3.10) and (3.13) show that $\mathcal{A}_{\sigma}=\mathcal{A}$.
Comparing relations (2.11) and (3.1) we see that the spectral function of
$\widetilde{\Theta}$ coincides with $S$. ∎

Theorem 3.1 provides characteristic properties for the model representation
$\mathcal{A}$ of the associated operator of a Jacobi-type pencil. It is seen
that these properties are close to the properties of the multiplication
operator $\Lambda_{0}$. Of course, $\Lambda_{0}$ itself satisfies properties
$(i)$–$(iii)$.

######  Corollary 3.3.

Let $\sigma$ be a non-negative measure on $\mathfrak{B}(\mathbb{R})$ with all
finite power moments, $\int_{\mathbb{R}}{\rm d}\sigma=1$,
$\int_{\mathbb{R}}|g(x)|^{2}{\rm d}\sigma>0$, for any non-zero complex
polynomial $g$. A linear operator $\mathcal{A}$ in $L^{2}_{\sigma}$ is a model
representation in $L^{2}_{\sigma}$ of the associated operator of a Jacobi-type
pencil if and only if properties $(i)$–$(iii)$ of Theorem 3.1 hold.

###### Proof.

It follows directly from our constructions in the proof of Theorem 3.1. ∎

Let $\Theta=(J_{3},J_{5},\alpha,\beta)$ be a Jacobi-type pencil and
$\mathcal{A}$ be a model representation in $L^{2}_{\sigma}$ of the associated
operator of $\Theta$. By the latter corollary we conclude that
$\mathcal{A}\Lambda_{0}$ is symmetric

$\displaystyle(\mathcal{A}\Lambda_{0}[u(\lambda)],[v(\lambda)])_{L^{2}_{\sigma}}=([u(\lambda)],\mathcal{A}\Lambda_{0}[v(\lambda)])_{L^{2}_{\sigma}},\qquad
u,v\in\mathbb{P}.$ (3.14)

Suppose that the orthogonality measure $\sigma$ is supported inside a finite
real segment $[a,b]$, $0<a<b<+\infty$, i.e.,
$\sigma(\mathbb{R}\backslash[a,b])=0$. In this case the operator $\Lambda$ of
the multiplication by an independent variable has a bounded inverse on the
whole $L^{2}_{\sigma}$. By (3.14) we may write

$\displaystyle\big{(}\Lambda^{-1}\mathcal{A}[\lambda u(\lambda)],[\lambda
v(\lambda)]\big{)}_{L^{2}_{\sigma}}=\big{(}\Lambda^{-1}[\lambda
u(\lambda)],\mathcal{A}[\lambda v(\lambda)]\big{)}_{L^{2}_{\sigma}},\qquad
u,v\in\mathbb{P}.$

Denote $\mathcal{P}_{0}=\Lambda\mathcal{P}$ and
$\mathcal{A}_{0}=\mathcal{A}|_{\mathcal{P}_{0}}$. Then

$\displaystyle\big{(}\Lambda^{-1}\mathcal{A}_{0}f,g\big{)}_{L^{2}_{\sigma}}=\big{(}\Lambda^{-1}f,\mathcal{A}_{0}g\big{)}_{L^{2}_{\sigma}},\qquad
f,g\in\mathcal{P}_{0}.$

Thus, in this case $\mathcal{A}_{0}$ is symmetric with respect to the form
$(\Lambda^{-1}\cdot,\cdot)_{L^{2}_{\sigma}}$. Analogous arguments were used in
the theory of operator pencils, see [7, Chapter IV, p. 163].

######  Example 3.4.

Let $a_{n}=1$, $b_{n}=c$, $c>2$, $\alpha_{n}\in\mathbb{R}$, $\beta_{n}=0$,
$\gamma_{n}=1$, $n\in\mathbb{Z}_{+}$. Define $J_{3}$ and $J_{5}$ by (1.2),
(1.3) with the above parameters. Consider a Jacobi-type pencil
$\Theta=(J_{3},J_{5},\alpha,\beta)$, with arbitrary $\alpha>0$ and
$\beta\in\mathbb{R}$. In this case

$\displaystyle r_{n}(x)=U_{n}\left(\frac{x-c}{2}\right),\qquad
n\in\mathbb{Z}_{+},$

where $U_{n}(t)=\frac{\sin((n+1)\arccos t)}{\sqrt{1-t^{2}}}$ is Chebyshev’s
polynomial of the second kind. The orthonormality relations for $r_{n}(x)$
have the following form

$\displaystyle\int_{c-2}^{c+2}r_{n}(\lambda)r_{m}(\lambda)\sqrt{1-\left(\frac{\lambda-c}{2}\right)^{2}}\frac{1}{\pi}{\rm
d}\lambda=\delta_{n,m},\qquad n,m\in\mathbb{Z}_{+}.$

By the recurrence relation (1.5) we calculate

$\displaystyle p_{2}(\lambda)=\alpha\lambda^{2}+(c+\beta)\lambda-\alpha_{0}.$

Since $(c+\beta)^{2}+4\alpha\alpha_{0}$ can be made zero or negative by a
proper choice of $\alpha_{0}$, then we have a non-classical case. Thus, this
case is worthy of an additional investigation which will be done elsewhere.

We can state the following moment problem for Jacobi-type pencils: find a non-
negative measure $\sigma$ on $\mathfrak{B}(\mathbb{R})$ with all finite power
moments, $\int_{\mathbb{R}}{\rm d}\sigma=1$, $\int_{\mathbb{R}}|g(x)|^{2}{\rm
d}\sigma>0$, for any non-zero complex polynomial $g$, and a linear operator
$\mathcal{A}$ in $L^{2}_{\sigma}$ with properties $(i)$–$(iii)$ of Theorem 3.1
such that

$\displaystyle\int_{\mathbb{R}}\mathcal{A}^{m}(1)\overline{\mathcal{A}^{n}(1)}{\rm
d}\sigma=s_{m,n},\qquad m,n\in\mathbb{Z}_{+},$

where $\\{s_{m,n}\\}_{m,n=0}^{\infty}$ is a prescribed set of real numbers
(called moments).

As usual, there appear three important questions: 1) the solvability of the
moment problem; 2) the uniqueness of a solution (the determinateness); 3) a
description of all solutions. This moment problem will be studied elsewhere.

##  4 A special perturbation of orthogonal polynomials
on a finite interval

If we look at the orthogonality relations (2.7) or at the representation of
the spectral function (2.11), we can see that we need to calculate a
polynomial of the operator $\mathcal{A}$. However, we do not have at hand any
functional calculus for the operator $\mathcal{A}$. It remains to calculate
the powers of $\mathcal{A}$ recurrently. It would be helpful to omit this
procedure and to have a more transparent relation. It is possible to do this
in the following special case.

Consider a Jacobi-type pencil $\Theta=(J_{3},J_{5},\alpha,\beta)$, with
$J_{3}$, $J_{5}$ defined by (1.2), (1.3), satisfying the following conditions

$\displaystyle\sigma(\mathbb{R}\backslash[-c,c])=0,\qquad 0<c<1,$ (4.1)
$\displaystyle
J_{5}=aJ_{3}^{2}+bJ_{3}+d\operatorname{diag}(1,0,0,0,\dots),\qquad a>0,\quad
b,d\in\mathbb{R},$
$\displaystyle\alpha=\frac{1}{aa_{0}},\qquad\beta=-\frac{b_{0}}{a_{0}}-\frac{b}{aa_{0}}.$
(4.2)

Here $\sigma$, as before (see the introduction), is the orthogonality measure
for polynomials $\\{r_{n}(x)\\}_{n=0}^{\infty}$ (related to $J_{3}$).

Introduce other related objects from the introduction: the space
$L^{2}_{\sigma}$, the operators $A$, $U$ and $\mathcal{A}=UAU^{-1}$. Observe
that

$\displaystyle UJ_{3}U^{-1}[p(x)]=[xp(x)],\qquad p\in\mathbb{P}.$

For an arbitrary $n\in\mathbb{Z}_{+}$ we may write

$\displaystyle\mathcal{A}[xr_{n}(x)]=UAU^{-1}UJ_{3}U^{-1}[r_{n}(x)]=UAJ_{3}\vec{e}_{n}=UJ_{5}\vec{e}_{n}$
$\displaystyle\hphantom{\mathcal{A}[xr_{n}(x)]}{}=U\big{(}aJ_{3}^{2}+bJ_{3}+d\operatorname{diag}(1,0,0,0,\dots)\big{)}\vec{e}_{n}$
$\displaystyle\hphantom{\mathcal{A}[xr_{n}(x)]}{}=aUJ_{3}U^{-1}UJ_{3}\vec{e}_{n}+bUJ_{3}\vec{e}_{n}+dU\operatorname{diag}(1,0,0,0,\dots)\vec{e}_{n}$
$\displaystyle\hphantom{\mathcal{A}[xr_{n}(x)]}{}=\big{[}(ax+b)xr_{n}(x)+d(r_{n},r_{0})_{L^{2}_{\sigma}}\big{]}.$

By the linearity we get

$\displaystyle\mathcal{A}[xp(x)]=\big{[}(ax+b)xp(x)+d(p,1)_{L^{2}_{\sigma}}\big{]},\qquad
p\in\mathbb{P}.$

Moreover, we have

$\displaystyle\mathcal{A}[1]=UAU^{-1}U\vec{e}_{0}=UA\vec{e}_{0}=U\frac{1}{\alpha}(\vec{e}_{1}-\beta\vec{e}_{0})=\left[\frac{1}{\alpha}(r_{1}(x)-\beta)\right].$

Choose an arbitrary polynomial $q(x)\in\mathbb{P}$. We may write
$q(x)=x\frac{q(x)-q(0)}{x}+q(0)$. Then

$\displaystyle\mathcal{A}[q(x)]=\mathcal{A}\left[x\frac{q(x)-q(0)}{x}\right]+\mathcal{A}[q(0)]$
$\displaystyle\hphantom{\mathcal{A}[q(x)]}{}=\left[\big{(}ax^{2}+bx\big{)}\frac{q(x)-q(0)}{x}+d\left(\frac{q(x)-q(0)}{x},1\right)_{L^{2}_{\sigma}}+\frac{q(0)}{\alpha}(r_{1}(x)-\beta)\right]$
$\displaystyle\hphantom{\mathcal{A}[q(x)]}{}=\left[(ax+b)q(x)+d\left(\frac{q(x)-q(0)}{x},1\right)_{L^{2}_{\sigma}}\right].$

Denote

$\displaystyle s_{k}:=\int_{\mathbb{R}}x^{k}{\rm d}\sigma,\qquad
k\in\mathbb{Z}_{+},\qquad s_{-1}:=0.$

Observe that

$\displaystyle\mathcal{A}\left[\sum_{k=0}^{\infty}c_{k}x^{k}\right]=\sum_{k=0}^{\infty}c_{k}\mathcal{A}\big{[}x^{k}\big{]}=\sum_{k=0}^{\infty}c_{k}\big{[}ax^{k+1}+bx^{k}+ds_{k-1}\big{]},\qquad
c_{k}\in\mathbb{C},$

where all but finite number of $c_{k}$ are zeros (i.e., $\mathcal{A}$ acts on
polynomials).

Observe that the operator $\mathcal{A}$ can be unbounded: the corresponding
example will be considered below. Thus, there appears a question: how to
simplify the calculation of $u(\mathcal{A})$ for a complex polynomial $u$ in
the integral representation (2.11)? Is there any (at least) polynomial
calculus for $\mathcal{A}$? The answer is affirmative. Consider the following
transformation from $L^{2}_{\sigma}$ to $l_{2}$

$\displaystyle
G\left[\sum_{k=0}^{\infty}c_{k}x^{k}\right]=\sum_{k=0}^{\infty}c_{k}\vec{e}_{k},\qquad
c_{k}\in\mathbb{C}.$

Here all but finite number of $c_{k}$ are zeros. This will be assumed in what
follows when dealing with operators on polynomials. Since $\sigma$ satisfies
conditions for the integral representation (3.1) (see the Necessity of the
proof of Theorem 3.1), then $\int_{\mathbb{R}}|g(x)|^{2}{\rm d}\sigma>0$, for
any non-zero complex polynomial $g$. This shows that two different polynomials
can not belong to the same class of the equivalence in $L^{2}_{\sigma}$. Thus,
the operator $G$ is well-defined. It is a linear operator with
$D(G)=\mathcal{P}$ and $R(G)=l_{2,{\rm f\/in}}$. Moreover, $G$ is invertible,
since the polynomial is uniquely determined by its coefficients. Set

$\displaystyle\mathbf{A}=G\mathcal{A}G^{-1}.$

Then $\mathbf{A}$ is a linear operator in $l_{2}$ with
$D(\mathbf{A})=l_{2,{\rm f\/in}}$. Consider the following shift operator on
the whole $l_{2}$

$\displaystyle Sx=\sum_{k=0}^{\infty}c_{k}\vec{e}_{k+1},\qquad x\in
l_{2},\quad x=(c_{k})_{k=0}^{\infty},\quad c_{k}\in\mathbb{C}.$

Notice that $S$ is linear and $\|Sx\|=\|x\|$, $\forall\,x\in l_{2}$. We may
write

$\displaystyle\mathbf{A}\left(\sum_{k=0}^{\infty}c_{k}\vec{e}_{k}\right)=(aS+bE)\sum_{k=0}^{\infty}c_{k}\vec{e}_{k}+\left(d\sum_{k=0}^{\infty}c_{k}s_{k-1}\right)\vec{e}_{0}.$

Here all but finite number of $c_{k}\in\mathbb{C}$ are zeros. This will be
assumed in the sequel for operators on $l_{2,{\rm f\/in}}$. By condition (4.1)
we see that

$\displaystyle|s_{n}|=\left|\int_{\mathbb{R}}x^{n}\chi_{[-c,c]}(x){\rm
d}\sigma\right|\leq\int_{\mathbb{R}}\big{|}x^{n}\chi_{[-c,c]}(x)\big{|}{\rm
d}\sigma\leq c^{n},\qquad n\in\mathbb{Z}_{+},$

where $\chi_{[-c,c]}(x)$ is the characteristic function of the segment
$[-c,c]$. Therefore $\vec{s}:=\sum\limits_{k=1}^{\infty}s_{k-1}\vec{e}_{k}$
belongs to $l_{2}$. Consequently, we may write

$\displaystyle\mathbf{A}w=(aS+bE)w+d(w,\vec{s})_{l_{2}}\vec{e}_{0},\qquad w\in
l_{2,{\rm f\/in}}.$

Set

$\displaystyle\widehat{\mathbf{A}}w=(aS+bE)w+d(w,\vec{s})_{l_{2}}\vec{e}_{0},\qquad
w\in l_{2}.$

The linear operator $\widehat{\mathbf{A}}$ is an extension of $\mathbf{A}$.
Observe that the operator $\widehat{\mathbf{A}}$ is bounded and

$\displaystyle\big{\|}\widehat{\mathbf{A}}\big{\|}\leq
a+|b|+|d|\|\vec{s}\|_{l_{2}}.$

We can apply Riesz’s calculus for $\widehat{\mathbf{A}}$. For an arbitrary
polynomial $u\in\mathbb{P}$ we may write

$\displaystyle
G\big{(}u\big{(}\mathcal{A}\big{)}[1]\big{)}=u\big{(}\widehat{\mathbf{A}}\big{)}\vec{e}_{0}=-\frac{1}{2\pi
i}\int_{\gamma}u(z)R_{z}\big{(}\widehat{\mathbf{A}}\big{)}{\rm
d}z\cdot\vec{e}_{0}=-\frac{1}{2\pi
i}\int_{\gamma}u(z)\big{(}R_{z}\big{(}\widehat{\mathbf{A}}\big{)}\vec{e}_{0}\big{)}{\rm
d}z,$ (4.3)

where $\gamma$ is a circle centered at zero with a radius $\rho$ bigger then
$a+|b|+|d|\|\vec{s}\|_{l_{2}}$, and
$R_{z}\big{(}\widehat{\mathbf{A}}\big{)}=\big{(}\widehat{\mathbf{A}}-zE\big{)}^{-1}$.
Here the last integral converges in the norm of $l_{2}$.

Let us calculate $R_{z}(\widehat{\mathbf{A}})\vec{e}_{0}=:\vec{f}$,
$z\in\gamma$. We may write

$\displaystyle\vec{e}_{0}=\big{(}\widehat{\mathbf{A}}-zE\big{)}\vec{f}=(aS+bE)\vec{f}+d\big{(}\vec{f},\vec{s}\big{)}_{l_{2}}\vec{e}_{0}-z\vec{f}.$

Then

$\displaystyle\frac{1}{a}\big{(}1-d\big{(}\vec{f},\vec{s}\big{)}_{l_{2}}\big{)}\vec{e}_{0}=\left(S-\left(\frac{z-b}{a}\right)E\right)\vec{f}.$

Since $z\in\gamma$, then $\big{|}\frac{z-b}{a}\big{|}>1$ and

$\displaystyle\vec{f}=\frac{1-d(\vec{f},\vec{s})_{l_{2}}}{a}\left(S-\left(\frac{z-b}{a}\right)E\right)^{-1}\vec{e}_{0}.$
(4.4)

Denote
$\vec{u}=\sum\limits_{k=0}^{\infty}u_{k}\vec{e}_{k}:=\big{(}S-\big{(}\frac{z-b}{a}\big{)}E\big{)}^{-1}\vec{e}_{0}$.
Then

$\displaystyle\vec{e}_{0}=\left(S-\left(\frac{z-b}{a}\right)E\right)\vec{u}=S\vec{u}+\left(\frac{b-z}{a}\right)\vec{u}.$

For the components of $\vec{u}$ we obtain the following equations

$\displaystyle\left(\frac{b-z}{a}\right)u_{0}=1,\qquad
u_{n-1}+\left(\frac{b-z}{a}\right)u_{n}=0,\qquad n\in\mathbb{N}.$

Then

$\displaystyle\vec{u}=\sum_{k=0}^{\infty}(-1)\left(\frac{a}{z-b}\right)^{k+1}\vec{e}_{k}.$

By (4.4) we may write

$\displaystyle\vec{f}=\tau\sum_{k=0}^{\infty}(-1)\left(\frac{a}{z-b}\right)^{k+1}\vec{e}_{k},\qquad\tau\in\mathbb{C},$

where

$\displaystyle\tau=\frac{1-d\big{(}\vec{f},\vec{s}\big{)}_{l_{2}}}{a}.$

Then

$\displaystyle
a\tau=1-d\left(\tau\sum_{k=0}^{\infty}(-1)\left(\frac{a}{z-b}\right)^{k+1}\vec{e}_{k},\vec{s}\right)_{l_{2}},$

and

$\displaystyle\tau=\frac{1}{a+d\sum\limits_{k=0}^{\infty}(-1)\big{(}\frac{a}{z-b}\big{)}^{k+1}s_{k-1}}.$

Denote

$\displaystyle
s(z):=\sum_{k=0}^{\infty}(-1)\left(\frac{a}{z-b}\right)^{k+1}s_{k-1},\qquad
z\in\gamma,$
$\displaystyle\vec{v}(z):=\sum_{k=0}^{\infty}(-1)\left(\frac{a}{z-b}\right)^{k+1}\vec{e}_{k},\qquad
z\in\gamma.$

Then

$\displaystyle
R_{z}\big{(}\widehat{\mathbf{A}}\big{)}\vec{e}_{0}=\vec{f}=\frac{1}{a+ds(z)}\vec{v}(z),\qquad
z\in\gamma.$

Thus, we obtained a transparent expression for
$R_{z}\big{(}\widehat{\mathbf{A}}\big{)}\vec{e}_{0}$. It can be used in
relation (4.3) and it gives a transparent expression for $u(\mathcal{A})[1]$.

######  Example 4.1.

Let $\Theta$ be a Jacobi-type pencil with $\alpha=\beta=\sqrt{2}$,
$a_{k}=\sqrt{2}$, $b_{k}=2$, $k\in\mathbb{Z}_{+}$, $\alpha_{n}=\beta_{n}=0$,
$\gamma_{n}=1$, $n\in\mathbb{Z}_{+}$, and $J_{3}$, $J_{5}$ have form (1.2),
(1.3). This Jacobi type pencil was considered in [16] and explicit formulas
for the associated polynomials $p_{n}(\lambda)$ were obtained.

Let $\kappa$ be an arbitrary positive number bigger than $2+2\sqrt{2}$.
Consider a Jacobi-type pencil
$\widehat{\Theta}=\big{(}\widehat{J}_{3},\widehat{J}_{5},\widehat{\alpha},\widehat{\beta}\big{)}$,
where

$\displaystyle\widehat{J}_{3}=\frac{1}{\kappa}J_{3},\qquad\widehat{J}_{5}=\frac{1}{\kappa}J_{5},\qquad\widehat{\alpha}=\alpha,\qquad\widehat{\beta}=\beta.$

Notice that the associated polynomials and the associated operators for
$\Theta$ and $\widehat{\Theta}$ are the same. It was shown in [16] that the
associated operator for $\Theta$ is unbounded. Thus, the associated operator
for the pencil $\widehat{\Theta}$ is unbounded, as well. The pencil
$\widehat{\Theta}$ satisfies conditions (4.1), (4.2) with

$\displaystyle c=\frac{2+2\sqrt{2}}{\kappa},\qquad a=\frac{1}{2}\kappa,\qquad
b=-2,\qquad d=\frac{1}{\kappa}.$

### Acknowledgements

The author is grateful to referees for their valuable comments and suggestions
which led to an essential improvement of the paper.

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