Nearly Invariant Subspaces for Operators in Hilbert Spaces
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Complex Analysis and Operator Theory
Nearly Invariant Subspaces for Operators in Hilbert Spaces Yuxia Liang1 · Jonathan R. Partington2 Received: 27 March 2020 / Accepted: 22 October 2020 © Springer Nature Switzerland AG 2020
Abstract For a shift operator T with finite multiplicity acting on a separable infinite dimensional Hilbert space we represent its nearly T −1 invariant subspaces in Hilbert space in terms of invariant subspaces under the backward shift. Going further, given any finite Blaschke product B, we give a description of the nearly TB−1 invariant subspaces for the operator TB of multiplication by B in a scale of Dirichlet-type spaces. Keywords Nearly invariant subspace · Shift operator · Blaschke product · Dirichlet-type space Mathematics Subject Classification Primary 47B38 · Secondary 47A15
1 Introduction Given α a real number, the Dirichlet-type space Dα (D) consists of all analytic functions k f (z) = ∞ k=0 ak z in D such that its norm f α :=
∞
1/2 |ak | (k + 1) 2
α
< +∞.
k=0
If α = −1, D−1 = A2 (D) the classical Bergman space, for α = 0, D0 = H 2 (D) the Hardy space, and for α = 1, D1 = D(D) the classical Dirichlet space, systemati-
Communicated by Isabelle Chalendar.
B
Yuxia Liang [email protected] Jonathan R. Partington [email protected]
1
School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, People’s Republic of China
2
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK 0123456789().: V,-vol
5
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Y. Liang, J. R. Partington
cally investigated in the book [6]. These are particular instances of separable infinite dimensional Hilbert spaces, to be denoted by H in this paper. We let B(H) denote the collection of all bounded linear operators acting on H. The notations N0 and N denote the set of all nonnegative integers and positive integers, respectively. Here we recall the Cl -vector-valued Hardy space H 2 (D, Cl ) consists of all analytic F : D → Cl such that the norm
1 F = sup 2π 0 0 : ∀ h ∈ H, chH ≤ uhH } ∈]0, ∞[.
(3.1)
Erard gave the definition of “nearly invariant under division by u”, which is same as “nearly Mu−1 invariant”, a special case of Definition 1.2. Considering the importance of the backward shift Mz , Erard proved the following theorem on nearly S ∗ invariant subspaces in H, under the assumption that Mz : H → H is bounded below. Theorem 3.1 [7, Theorem 5.1] Assume that H satisfies (i)-(iv) with u(z) = z, and dim(H Mz H) = 1 and hH ≤ Mz hH for all h ∈ H. Assume also that there exists f ∈ H with f (0) = 0. Let M be a nonzero subspace of H which is nearly invariant under the backward shift Mz . Let g be any unit vector of M (M ∩ Mz H). Then there exists a linear submanifold N of H 2 (D) such that M = gN and for all h ∈ M, we have h hH ≥ g 2 . H (D) Besides, N is invariant under the backward shift and g(0) = 0. We note the operator TB : Dα → Dα is more general than Mz : H 2 (D) → H 2 (D), so we seek characterizations for nearly TB−1 invariant subspaces in Dα with α ∈
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