Tridiagonal shifts as compact + isometry

DOI10.1007/S00013-022-01780-8arXiv2111.04180MaRDI QIDQ6382426

Publication date: 7 November 2021

Abstract: Let

{a_{n}}_{n g e q 0}

and

{b_{n}}_{n g e q 0}

be sequences of scalars. Suppose

a_{n} e q 0

for all

n g e q 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

m a t h b b D = z i n m a t h b b 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

| f r a c b_{n} a_{n} - f r a c b_{n + 1} a_{n + 1} | i g h t a r r o w 0

and

| f r a c a_{n} a_{n + 1} | i g h t a r r o w 1

.

Mathematics Subject Classification ID

Linear operators defined by compactness properties (47B07) Perturbation theory of linear operators (47A55) Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) (47B37) Linear operators on function spaces (general) (47B38) Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) (46E22) Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) (47B32) Hardy spaces (30H10)

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