Almost Invariant Subspaces of the Shift Operator on Vector-Valued Hardy Spaces
- PDF / 376,808 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 102 Downloads / 163 Views
Integral Equations and Operator Theory
Almost Invariant Subspaces of the Shift Operator on Vector-Valued Hardy Spaces Arup Chattopadhyay, Soma Das and Chandan Pradhan Abstract. In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vectorvalued Hardy space which is a vectorial generalization of a result of Chalendar–Gallardo–Partington. Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space. Mathematics Subject Classification. 47A13, 47A15, 47A80, 46E20, 47B38, 47B32, 30H10. Keywords. Vector valued Hardy space, Nearly invariant subspaces, Almost invariant subspaces, Shift operator, Beurling’s theorem, Half space, Multiplier operator.
1. Introduction In 1988, Hitt [9] first introduces the notion of nearly invariant subspaces under the backward shift operator acting on the scalar-valued Hardy space which he used as a tool for classifying the simply shift-invariant subspaces of the Hardy space of an annulus. In his paper he rather called it as “weakly invariant subspace under the backward shift ”. Later Sarason [14] further investigated these spaces and modified Hitt’s algorithm for scalar-valued Hardy space to study the kernels of Toeplitz operators. In 2010, Chalendar–Chevrot–Partington (C-C-P) [4] gives a complete characterization of nearly invariant subspaces under the backward shift operator acting on the vector-valued Hardy space, providing a vectorial generalization of a result of Hitt. Recently Chalendar– Gallardo–Partington (C–G–P) [5] introduce the notion of nearly invariant subspace of finite defect for the backward shift operator acting on the scalar valued Hardy space as a generalization of nearly invariant subspaces and This paper is published in the form submitted to IEOT on January 14, 2020. A related paper by R. O’Loughlin was submitted to arxiv some months later, see arXiv:2005.00378 [math.FA]. 0123456789().: V,-vol
52
Page 2 of 15
A. Chattopadhyay et al.
IEOT
provides a complete characterization of these spaces in terms of backward shift invariant subspaces. Using this characterization they also described the almost-invariant subspaces for the shift and its adjoint acting on the scalar valued Hardy space. In this connection we should mention that the relation between nearly invariant subspaces under the backward shift and the kernel of Toeplitz operators has been discussed in [6]. In this paper we further study nearly invariant subspaces of finite defect under the backward shift operator acting on the vector valued Hardy space and provides a vectorial generalization of C–G–P algorithm. As a consequences we completely characterize nearly invariant subspaces of finite defect under the backward shift in terms of backward shift invariant subspaces. Furthermore, using the characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant s
Data Loading...
