Toeplitz operators (or equivalently, Wiener-Hopf operators; more generally, block Toeplitz operators; and particularly, Toeplitz determinants) are of importance in connection with a variety of problems in physics, and in particular, in the field of quantum mechanics. For example, a study of solvable models in quantum mechanics uses the spectral theory of Toeplitz operators (cf. [Pr]); the one-d...

In 1950, P. R. Halmos, motivated in part by the successful development of the theory of normal operators, introduced the notions of subnormality and hyponormality for (bounded) Hilbert space operators. An operator T is subnormal if it is the restriction of a normal operator to an invariant subspace; T is hyponormal if T*T > TT*. It is a simple matrix calculation to verify that subnormality impl...

The concept of K-quasi-hyponormal operators on semi-Hilbertian space is defined by Ould Ahmed Mahmoud Sid and Abdelkader Benali in [7]. This paper devoted to the study new class H, ∥. ∥Acalled (n,m)power-A-quasi-hyponormal denoted [(n,m)QH]A.We give some basic properties these examples are also given .An operator T ∈ BA(H) said be (n,m) power-A-quasi-hyponormal for positive A integers n m if T⋕...

We are concerned with fractional powers of the so-called hyponormal operators of Putnam type. Under some suitable assumptions it is shown that if A, B are closed hyponormal linear operators of Putnam type acting on a complex Hilbert space H, then D((A+B)α) = D(Aα)∩D(Bα) = D((A+B)∗α) for each α ∈ (0,1). As an application, a large class of the Schrödinger’s operator with a complex potential Q ∈ L...

In this paper we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space H Cn of the unit circle. Firstly, we establish a tractable and explicit criterion on the hyponormality of block Toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator. Secondly, we consider the gap...

We make some remarks concerning the invariant subspace problem for hyponormal operators. In particular, we bring together various hypotheses that must hold for a hyponormal operator without nontrivial invariant subspaces, and we discuss the existence of such operators. 2000 Mathematics Subject Classification. 47B20, 47A15. Let be a separable, infinite-dimensional, complex Hilbert space and deno...

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 ...

A survey of the theory of k-hyponormal operators starts with the construction of a polynomially hyponormal operator which is not subnormal. This is achieved via a natural dictionary between positive functionals on specific convex cones of polynomials and linear bounded operators acting on a Hilbert space, with a distinguished cyclic vector. The class of unilateral weighted shifts provides an op...