We introduce a new class of operator algebras on Hilbert space. To each bounded linear operator a spectral algebra is associated. These algebras are quite substantial, each containing the commutant of the associated operator, frequently as a proper subalgebra. We establish several sufficient conditions for a spectral algebra to have a nontrivial invariant subspace. When the associated operator ...

In this paper we prove that if S equals a finite sum of finite products of Toeplitz operators on the Bergman space of the unit disk, then S is compact if and only if the Berezin transform of S equals 0 on ∂D. This result is new even when S equals a single Toeplitz operator. Our main result can be used to prove, via a unified approach, several previously known results about compact Toeplitz oper...

In previous work, we have developped methods to compute norms and spectral radii of transition operators on proper metric spaces. The operators are assumed to be invariant under a locally compact, amenable group which acts with compact quotient. Here, we present several further applications of those methods. The first concerns a generalization of an identity of Hardy, Littlewood and Pólya. The ...

This thesis deals with linear ill-posed problems related to compact operators, and iterative Krylov subspace methods for solving discretized versions of these. Linear compact operators in infinite dimensional Hilbert spaces will be investigated and several results on the singular values and eigenvalues for such will be presented. A large subset of linear compact operators consists of integral o...

In this paper we consider simultaneously triangularizable collections of compact operators and show that similarities of any finite subcollection can be made arbitrarily close to commuting normal operators. As a consequence, we obtain a variant of a theorem of G.-C. Rota. Also, we give some sufficient conditions for simultaneous triangularization of collections of compact operators. Finally, se...

We prove that a Banach space E has the compact range property (CRP) if and only if for any given C∗-algebra A, every absolutely summing operator from A into E is compact. Related results for p-summing operators (0 < p < 1) are also discussed as well as operators on non-commutative L-spaces and C∗-summing operators.