Inner sequence based invariant subspaces in $H^{2}(D^2)$

Abstract

Abstract.A closed subspace H 2 (D 2 ) is said to be invariant if it is invariant under the Toeplitz operators T z and T w . Invariant subspaces of H 2 (D 2 ) are well-known to be very complicated. So discovering some good examples of invariant subspaces will be beneficial to the general study. This paper studies a type of invariant subspace constructed through a sequence of inner functions. It will be shown that this type of invariant subspace has direct connections with the Jordan operator. Related calculations also give rise to a simple upper bound for j 1 − |λ j |, where {λ j } are zeros of a Blaschke product.

Two Inner Sequences Based Invariant Subspaces in $${{H}^{2} (\mathbb{D}^{2})}$$ H 2 ( D 2 )

“…w j is called the inner based invariant subspaces by Seto and Yang in [13]. The inner based invariant subspaces have been very well studied in [11][12][13], and in particular, when ϕ 0 (z) is a Blaschke product, Izuchis (cf.…”

“…The inner based invariant subspaces have been very well studied in [11][12][13], and in particular, when ϕ 0 (z) is a Blaschke product, Izuchis (cf. [6]) determined the rank of M completely.…”

“…In general, it is very hard to determine these two functions. Recently, [11] determined these two functions for two well-known invariant subspaces [8,12,13] in H 2 (D 2 ) and they turn out to be strikingly simple. We will focus on the invariant subspace given by the following operatorvalued inner function.…”

A Perturbation Theory for Core Operators of Hilbert-Schmidt Submodules

“…For the detail of these submodules, see [15] and [16]. We define k subsets of integers for each inner sequence {q j } ∞ j=0 and λ 1 in D as follows:…”

Two Types of Invariant Subspaces in the Polydisc

“…This question was asked by Rudin in his book [4, p.78] and it is still open. Recently, for n = 2, two types of important invariant subspaces known as inner-sequence based invariant subspaces and invariant subspaces generated by two inner functions have been extensively studied by various authors in different context(see [6,9,8,7,11,12]). In this paper, inspired from these studies, we define two new types of invariant subspaces of H 2 (D n ) by considering a larger class of functions than inner functions.…”