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...

For Hilbert space operators A and B, let δAB denote the generalised derivation δAB(X) = AX − XB and let 4AB denote the elementary operator4AB(X) = AXB−X. If A is a pk-quasihyponormal operator, A ∈ pk−QH, and B∗ is an either p-hyponormal or injective dominant or injective pk−QH operator (resp., B∗ is an either p-hyponormal or dominant or pk −QH operator), then δAB(X) = 0 =⇒ δA∗B∗(X) = 0 (resp., ...

In this paper, we show that for a hyponormal operator the analytic spectral subspace coincides with the algebraic spectral subspace. Using this result, we have the following result: Let T be a operator with property (δ) on a Banach space X and let S be a hyponormal operator on a Hilbert space H. Then every generalized intertwining linear operator θ : X → H for (S, T ) is continuous if and only ...

It is well known that for multipliers f of the Drury-Arveson space H n, ‖f‖∞ does not dominate the operator norm of Mf . We show that in general ‖f‖∞ does not even dominate the essential norm of Mf . A consequence of this is that there exist multipliers f of H n for which Mf fails to be essentially hyponormal, i.e., if K is any compact, self-adjoint operator, then the inequality M∗ f Mf −MfM f ...

Let A be a bounded linear operator acting on infinite dimensional separable Hilbert space H. Browder’s theorem and a-Browder’s theorem are related to an important property which has a leading role on local spectral theory: the single valued extension property see the recent monograph by Laursen and Neumann [19]. The study of operators satisfying Browder’s theorem is of significant interest, and...

In this article we develop a general method by which one can explicitly evaluate certain sums of nth powers products d≥1 elementary trigonometric functions evaluated at m=(m1,…,md)-th roots unity. Our approach is to first identify the individual terms in expression under consideration as eigenvalues discrete Laplace operator associated graph whose vertices form d-dimensional torus Gm depends on...

We prove an n-tuple analogue of the Putnam area inequality for the spectrum of a single p-hyponormal operator. Let B…H† denote the algebra of operators (i.e. bounded linear transformations) on a separable Hilbert space H. The operator A 2 B…H† is said to be p-hyponormal, 0 < p 1, if jA j2p jAj2p. Let H…p† denote the class of p-hyponormal operators. Then H…1† consists of the class of p-hyponorma...

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...