k-HYPONORMALITY OF POWERS OF WEIGHTED SHIFTS VIA SCHUR PRODUCTS
نویسنده
چکیده
Let H be a separable, infinite dimensional complex Hilbert space and let B(H) be the algebra of bounded linear operators on H. An operator T∈ B(H) is said to be normal if T ∗T = TT ∗, subnormal if T is the restriction of a normal operator (acting on a Hilbert space K ⊇ H) to an invariant subspace, and hyponormal if T ∗T ≥ TT ∗. The Bram-Halmos criterion for subnormality states that an operator is subnormal if and only if
منابع مشابه
When Is Hyponormality for 2-variable Weighted Shifts Invariant under Powers?
Abstract. For 2-variable weighted shifts W(α,β) ≡ (T1, T2) we study the invariance of (joint) khyponormality under the action (h, `) 7→ W (h,`) (α,β) := (T h 1 , T ` 2 ) (h, ` ≥ 1). We show that for every k ≥ 1 there exists W(α,β) such that W (h,`) (α,β) is k-hyponormal (all h ≥ 2, ` ≥ 1) but W(α,β) is not k-hyponormal. On the positive side, for a class of 2-variable weighted shifts with tensor...
متن کاملk-HYPONORMALITY OF FINITE RANK PERTURBATIONS OF UNILATERAL WEIGHTED SHIFTS
Abstract. In this paper we explore finite rank perturbations of unilateral weighted shifts Wα. First, we prove that the subnormality of Wα is never stable under nonzero finite rank pertrubations unless the perturbation occurs at the zeroth weight. Second, we establish that 2-hyponormality implies positive quadratic hyponormality, in the sense that the Maclaurin coefficients ofDn(s) := detPn [(W...
متن کاملHyponormality and Subnormality for Powers of Commuting Pairs of Subnormal Operators
Let H0 (resp. H∞) denote the class of commuting pairs of subnormal operators on Hilbert space (resp. subnormal pairs), and for an integer k ≥ 1 let Hk denote the class of k-hyponormal pairs in H0. We study the hyponormality and subnormality of powers of pairs in Hk. We first show that if (T1, T2) ∈ H1, the pair (T 2 1 , T2) may fail to be in H1. Conversely, we find a pair (T1, T2) ∈ H0 such tha...
متن کاملAluthge transforms of 2-variable weighted shifts
We introduce two natural notions of multivariable Aluthge transforms (toral and spherical), and study their basic properties. In the case of 2-variable weighted shifts, we first prove that the toral Aluthge transform does not preserve (joint) hyponormality, in sharp contrast with the 1-variable case. Second, we identify a large class of 2-variable weighted shifts for which hyponormality is pres...
متن کاملk-HYPONORMALITY OF MULTIVARIABLE WEIGHTED SHIFTS
We characterize joint k-hyponormality for 2-variable weighted shifts. Using this characterization we construct a family of examples which establishes and illustrates the gap between k-hyponormality and (k+1)-hyponormality for each k ≥ 1. As a consequence, we obtain an abstract solution to the Lifting Problem for Commuting Subnormals. 1. Notation and Preliminaries The Lifting Problem for Commuti...
متن کامل