This paper discusses several structural and fundamental properties of the $k^{th}$-order slant Toeplitz operators on Lebesgue space $n$- torus $\mathbb{T}^n$, for integers $k\geq 2$ $n\geq 1$. We obtain certain equivalent conditions commutativity essential these operators. In last section, we deal with spectrum a operator $L^2(\mathbb{T}^n)$ investigate such an to be isometry, hyponormal or nor...

In this note we prove that if A and B∗ are subnormal operators and X is a bounded linear operator such that AX − XB is a Hilbert-Schmidt operator, then f(A)X −Xf(B) is also a Hilbert-Schmidt operator and ||f(A)X −Xf(B)||2 ≤ L ||AX −XB||2, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X ∈ L(H) i...

We prove that if A and B∗ are subnormal operators and X is a bounded linear operator such that AX −XB is a Hilbert-Schmidt operator, then f A X −Xf B is also a Hilbert-Schmidt operator and ‖f A X −Xf B ‖2 ≤ L‖AX −XB‖2 for f belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X ∈ L H is such that SX − XT belo...

In this paper we deal with the subnormality and the quasinormality of Toeplitz operators with matrix-valued rational symbols. In particular, in view of Halmos’s Problem 5, we focus on the question: Which subnormal Toeplitz operators are normal or analytic ? We first prove: Let Φ ∈ L∞Mn be a matrix-valued rational function having a “matrix pole,” i.e., there exists α ∈ D for which kerHΦ ⊆ (z−α)H...

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...

We characterize hyponormal “rational” Toeplitz pairs which are pairs of Toeplitz operators whose symbols are rational functions in L∞. The main result of this article is as follows. If T = (Tφ, Tψ) is a hyponormal rational Toeplitz pair then φ− βψ ∈ H for some constant β; in other words, their co-analytic parts necessarily coincide up to a constant multiple. As a corollary we get a complete cha...

Let B(H) be the set of all bounded endomorphisms (operators) on the complex Hubert space H. T € B(H) is paranormal if \\(T — z/) || = l/d(z, cr(T)) for all z £ cr(T) where d(z, cr(T)) is the distance from z to cr(T), the spectrum of T. If J is the set of all paranormal operators on H, then J contains the normal operators, JÍ, and the hyponormal operators; and J is contained in JL, the set of al...

Abstract We show that if a densely defined closable operator A is such the resolvent set of 2 nonempty, then necessarily closed. This result extended to case polynomial . also generalize recent by Sebestyén–Tarcsay concerning converse J. von Neumann. Other interesting consequences are given. One them proof T quasinormal (unbounded) normal for some , normal. Hence closed subnormal itself hyponor...