Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tridiagonal shifts as compact + isometry   Let $\{a_n\}_{n\geq 0}$ and $\{b_n\}_{n\geq 0}$ be sequences of scalars.
Suppose $a_n \neq 0$ for all $n \geq 0$. We consider the tridiagonal kernel
(also known as band kernel with bandwidth one) as \[ k(z, w) =
\sum_{n=0}^\infty ((a_n + b_n z)z^n) \overline{(({a}_n + {b}_n {w}) {w}^n)}
\qquad (z, w \in \mathbb{D}), \] where $\mathbb{D} = \{z \in \mathbb{C}: |z| <
1\}$. Denote by $M_z$ the multiplication operator on the reproducing kernel
Hilbert space corresponding to the kernel $k$. Assume that $M_z$ is
left-invertible. We prove that $M_z =$ compact $+$ isometry if and only if
$|\frac{b_n}{a_n}-\frac{b_{n+1}}{a_{n+1}}|\rightarrow 0$ and
$|\frac{a_n}{a_{n+1}}| \rightarrow 1$.

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The categories of this paper are: stat
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th