Small compact perturbation of strongly irreducible operators

An operatorT onH is called strongly irreducible ifT is not similar to any reducible operators. In this paper, we shall say yes to answer the following question raised by D. A. Herrero.Given an operatorT with connected spectrum (T) and a positive number , can we find a compact operatorK with K < such thatT+K is strongly irreducible?Supported by National Natural Science Foundation of China(19901011), Mathematical Center of State Education Commission of China and 973 Project of China     

The strongly irreducible operators in nest algebras

An operatorT on is called strongly irreducible ifT does not commute with any nontrivial idempotent operator. In this paper, we first show that each nest algebra ( ) has strongly irreducible operators. Secondly, we obtain a characterization of operators which can be uniquely written as a direct sum of finitely many strongly irreducible operators. Finally, we characterize the strongly irreducibility of operators in a nest algebra ( ).This project was partially supported by National Natural Science Foundation of China.     

ON THE SPECTRAL RADIUS OF IRREDUCIBLE AND WEAKLY IRREDUCIBLE OPERATORS IN BANACH LATPICES

Abstract

If T is an operator on a Banach lattice E we call T weakly irreducible if E contains no non-trivial T-invariant bands. We prove that if E is order complete and if the weakly irreducible operator T > 0 is in (E′_oo ? E)^⊥⊥ then T has positive spectral radéus. Prom this follows that Jentesch's theorem holds in arbitrary Banach function spaces.

If [Ttilde] denotes the restriction of T′ to E′_oo, 0 ? T an order continuous operator, then T is weakly irreducible if and only if [Ttilde]: E′_oo→E′_oo is weakly irreducible.

Finally we show that the majorizing, irreducible operator T ≥ 0, has positive spectral radius if either Tⁿ is weakly compact or E has property (P) or T is strongly majorizing.     

Factorization along commutative subspace lattices

A positive invertible operatorT is said to be factorable along a commutative subspace latticeL if there is an invertible operatorA inAlg L whose inverse is also inAlg L and such thatT=A*A. We investigate a number of conditions that are equivalent to factorability of a given operator along a latticeL. As a byproduct, we derive a condition that guarantees that the latticeT L, defined as {range(TE) E L} is commutative. Applications are suggested to the particular case of factoringL functions via analytic Toeplitz operators on the polydisc.     

On the commuting extensions of subnormal operators

We give some results concerning the following problem: Given a linear bounded operatorA which is subnormal on a Hilbert spaceH, andB its minimal normal extension on a Hilbert spaceK ～H, when can a quasi-normal operatorT commuting withA be extended to an operatorT ^e onK such thatT ^e commutes withB andT ^e is quasi-normal onK?     

Totally P-posinormal operators are subscalar

LetH be a complex infinite-dimensional separable Hilbert space. An operatorT inL(H) is called totally P-posinormal (see [9]) iff there is a polynomialP with zero constant term such that for each , whereT _z =T–zI andM(z) is bounded on the compacts of C. In this paper we prove that every totally P-posinormal operator is subscalar, i.e. it is the restriction of a generalized scalar operator to an invariant subspace. Further, a list of some important corollaries about Bishop's property and the existence of invariant subspaces is presented.     

On the properties of a class of integral operators in the space Lp

In the spaceL _p (?),p > 1, we consider the operatorA=a? +bS? +cP? +T?, wherea(t), b(t), and c(t) are piecewise-continuous functions on the contour ?, T is a completely continuous operator, P?=1/2πi∫ ?(τ) dτ/? τ ?t ? 1, S?=1/gpi∫ ?(τ) dτ/? τ ? i, ? is a closed convex Lyapunov contour having no rectilinear portions. We study the properties of the operator P and we show that the Noether property conditions and the index of the operator A do not depend on the term cP.     

A note on the theorems of M.G. Krein and L.A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle

Continuous analogs of orthogonal polynomials on the circle are solutions of a canonical system of differential equations, introduced and studied by Krein and recently generalized to matrix systems by Sakhnovich. We prove that the continuous analogs of the adjoint polynomials converge in the upper half-plane in the case of L² coefficients, but in general the limit can be defined only up to a constant multiple even when the coefficients are in L^p for any p>2, the spectral measure is absolutely continuous and the Szegö-Kolmogorov-Krein condition is satisfied. Thus, we point out that Krein's and Sakhnovich's papers contain an inaccuracy, which does not undermine known implications from these results.     

Finiteness of the discrete spectrum of some block Toeplitz operators

Sufficient conditions are given for the finiteness of the discrete spectrum of the block Toeplitz operatorT _A generated in the spaceH ₂ ⁿ by self-adjoint matrix functionA(t)(|t|=1). These results are obtained by means of theorems concerning the spectrum of a perturbed self-adjoint operators.     

Normal operators and multipliers on complex Banach spaces and a symmetry property ofL 1-predual spaces

An operatorT on a complex Banach spaceX is callednormal if there exists an operatorS such that (1/2)(T+S) and (1/2i)(T?S) are hermitian, andTS=ST. We show thatT: X→X is normal iffT′: X ^′ » X^′ is normal. Using a generalization of the principle of local reflexivity this result enables us to prove that multipliers on complexL ¹-predual spaces are always normal.