\}$ in ${\cal H}$ such that both $T_{\bm{e}}$ and $T_{\bm{e}}^{-1}$ are bounded, and $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a pair of semi-Riesz bases if there exists an ONB $\bm{e}= \{ e_{n} \}$ in ${\cal H}$ such that either $T_{\bm{e}}$ or $T_{\bm{e}}^{-1}$ are bounded. In this paper we consider the following operators in ${\cal H}$ defined by a sequence $\{ \phi_{n} \}$ in ${\cal H}$ and an ONB $\bm{e}= \{ e_{n} \}$ in ${\cal H}$: $$\begin{aligned}
T_{\phi, \bm{e}} 
&\equiv& \sum_{k=0}^{\infty} \phi_{k} \otimes \bar{e}_{k} , \nonumber \\
T_{\bm{e},\phi}
&\equiv& \sum_{k=0}^{\infty} e_{k} \otimes \bar{\phi}_{k} , \nonumber\end{aligned}$$ where the tensor $x \otimes \bar{y}$ of elements $x, \; y$ of ${\cal H}$ is defined by $$\begin{aligned}
(x \otimes \bar{y} ) \xi = (\xi | y)x, \;\;\; \xi \in {\cal H}. \nonumber\end{aligned}$$ This is also denoted by the Dirac notation $|x >< y|$. Here we use the notation $x \otimes \bar{y}$.

In Section 2, we investigate the relationship between the operator $T_{\bm{e}}$ and the operators $T_{\phi, \bm{e}}$ and $T_{\bm{e},\phi}$. The operator $T_{\bm{e},\phi}$ is always closed, however $D(T_{\phi,\bm{e}}^{\ast})$ is not necessarily dense in ${\cal H}$, equivalently, $T_{\bm{e}}$ and $T_{\phi,\bm{e}}$ are not necessarily closable. Indeed, it is shown that the following statements are equivalent:

