Taylor spectrum of commuting n-contractions

Abstract

In this paper, we characterize the Taylor spectrum for a certain class of commuting n-contractions. We also investigate the behavior of this spectrum under action of involutive automorphisms of the unit ball 𝔹 _n.     

UBIQUITOUS PROPERTIES OF CERTAIN OPERATIONAL QUANTITIES OF LINEAR RELATIONS IN NORMED SPACES

Abstract

Let X and Y be normed spaces and T a linear relation in X x Y. Certain known operational quantities of linear operators are extended to linear relations. Properties of T such as precompactness, strict singularity, upper semi-Fredholmness and partial continuity are defined and characterised in terms of the given quantities, and certain uniform boundedness properties of these quantities derived.     

JOINT SPECTRAL SETS

Abstract

We consider the problem: if S,TEB(H) are commuting operators with von Neumann spectral sets X and Y respectively, does it imply that X x Y is a joint spectral set for the pair (S,T)?     

w-Hyponormal Operators are Subscalar

We prove that if X is a Banach space, R, S B(X), then RS is subscalar (subdecomposable) if and only if SR is. As corollaries, it is shown that w-hyponormal operators (including p-hyponormal (p > 0) and log-hyponormal operators) and their Aluthge transformations and inverse Aluthge transformations are subscalar.     

Abstract,weighted, and multidimensional Adamjan-Arov-Krein theorems,and the singular numbers of Sarason commutants

The classical Adamjan-Arov-Krein (A-A-K) theorem relating the singular numbers of Hankel operators to best approximations of their symbols by rational functions is given an abstract version. This provides results for Hankel operators acting in weightedH ²(T; ), as well as inH ²(T ^d ), and an A-A-K type extension of Sarason's interpolation theorem. In particular, it is shown that all compact Hankel operators inH ²(T ^d ) are zero.Author partially supported by NSF grant DMS89-11717.     

DISJOINTNESS OF OPERATOR RANGES IN BANACH SPACES

Abstract

Let X be a Banach space. A linear subspace of X is called an operator range if it coincides with the range of a bounded linear operator defined on some Banach space. The paper studies disjointness and inclusion properties of various types of operator ranges in a separable infinite dimensional Banach space X. One of the main results is the following: Let E be a non-closed operator range in X. Then X contains a non-closed dense operator range R with the properties E∩= {0}, and R is decomposable, i.e. R = M + N where M,N are closed and infinite dimensional and M ∩ N = {0} (Theorem 6.2).     

r-ASYMPTOTICALLY QUASI-FINITE RANK OPERATORS AND THE SPECTRUM OF MEASURES

Abstract

The r-asymptotically quasi finite rank operators were introduced in [10]. For regular operators on Banach lattices, these operators are the order theoretic analogue of Riesz operators on Banach spaces. We establish their basic properties and apply these in the spectral analysis of convolution operators.     

Tangential interpolation with symmetries and two-point interpolation problem for matrix-valued H2-functions

We describe all solutions of the two-sided tangential interpolation problem in the class of matrix-valued Hardy functions when symmetries are added: these symmetries are defined in terms of involutions ofH ₂. The obtained results are applied to a one-sided two-points tangential interpolation for matrix functions.The research of this author is partially supported by the NSF Grant DMS 9500924 and by the Binational United States-Israel Foundation Grant 9400271.     

FACTORIZATION OF UNBOUNDED THIN AND COTHIN OPERATORS

Abstract

Let X and Y be normed spaces and T: D(T) ? X → Y a linear operator. Following R.D. Neidingcr [N1] we recall the Davis, Figiel, Johnson, Pelczynski factorization of T corresponding to a parameter p (1 ≤ p ≤ ∞) and apply the corresponding factorization result in [N1] to unbounded thin operators. Properties equivalent to ubiquitous thinness arc derived. Defining an operator T to be cothin if its adjoint is thin, a dual factorization result for cothin operators is obtained, where for each 1 < p < ∞, the intermediate space in the factorization is cohereditarily l_p. This result is shown to hold more generally for the cases when T is either partially continuous or closable; in particular, such operators are strictly cosingular. A condition for a closable weakly compact operator to be strictly cosingular follows as a corollary.     

ON ORBITS OF POLYNOMIAL MAPS IN BANACH SPACES

Abstract

We obtain some results concerning the notions of hypercyclicity and supercyclicity for continuous homogeneous polynomial maps in Banach spaces. We also present a generalization of a theorem due to Beauzamy concerning the existence of orbits tending to infinity for continuous linear operators with spectral radius > 1.     

CHARACTERISATIONS OF OPERATORS OF LOWER SEMI-FREDHOLM TYPE IN NORMED LINEAR SPACES

Abstract

Let X and Y be normed linear spaces. A linear operator T: D(T) ? X → Y is called an F-operator if its adjoint T′: D(T) ? Y′ → D(T)' is a φ+ -operator, i.e. has closed range and finite dimensional-kernel. Characterisations of an F_-operator T are obtained in the general case and in the case when T is closable. Unbounded strictly cosingular operators are defined and shown to belong to the class of F_ -admissible pertubations whenever Y is complete.     

Radius preserving (semi)regularities in Banach algebras

Abstract

In this note we investigate regularities (semiregularities) R and S in a Banach algebra A satisfying S ? R and the corresponding spectra σ_S and σ_R Satisfying

sup {|λ| : λ ∈ σ_R(a)} = sup{|λ| : λ ∈ σ_S (a)}     

Contractions Satisfying the Absolute Value Property

Let B(H) denote the algebra of operators on a complex Hilbert space H, and let U denote the class of operators which satisfy the absolute value condition . It is proved that if is a contraction, then either A has a nontrivial invariant subspace or A is a proper contraction and the nonnegative operator is strongly stable. A Putnam-Fuglede type commutativity theorem is proved for contractions A in , and it is shown that if normal subspaces of . It is proved that if are reducing, then every compact operator in the intersection of the weak closure of the range of the derivation with the commutant of A* is quasinilpotent.     

SOME SPECTRAL PROPERTIES OF MULTIPLIERS ON SEMI-PRIME BANACH ALGEBRAS

Abstract

We extend to arbitrary semi-prime Banach algebras some results of spectral theory and Fredholm theory obtained in [1] and [2] for multipliers defined in commutative semi-simple Banach algebras.     

ISOLATED POINTS OF THE APPROXIMATE POINT SPECTRUM OF CERTAIN LATTICE HOMOMORPHISMS ON Co (X)

Abstract

Suppose X is a locally compact Hausdorff space and C (X) the apace of all continuous complex valued functions on X which vanish at infinity. Let T be a (complex) linear lattice homomorphism on C_o (X) whose adjoint is also a lattice homomorphism. It is sham that every non-zero isolated point of the approximate point spectrum of T lies in the point spectrum of T. An example is given to show that the exclusion of zero is necessary, even when X is compact. The same techniques are then used to show that if also the spectrum of T is finite then T can be written, in a natural manner, as a direct sum of two such lattice homomorphisms; one being an n'th root of an invertible multiplication operator and the other quasi-nilpotent.     

Idempotents of Fourier multiplier algebra

We prove that if the indicator-function1 _E of a measurable setE is a Fourier multiplier in the spaceE ^p () for somep2 thenE is an open set (up to a set of measure zero).     

THE o-SPECTRUM OF r-ASYMPTOTICALLY QUASI-FINITE-RANK OPERATORS

Abstract

We introduce the class of r-asymptotically quasi finite rank operators (which properly contains the class of r-compact operators) and show that for these operators the spectrum and o-spectrum coincide.     

FUNCTIONAL CALCULI FOR HILBERT SPACE OPERATORS

Abstract

Let T be a bounded operator on a Hilbert space H with Von Neumann spectral set X. If there exists no non-zero reducing subspace of H restricted to which T is a normal operator with spectrum contained in the boundary of X and if the uniform algebra R(X) is pointwise boundedly dense in H^∞ (X°), then there exists a functional calculus f → f(T) for f ε H^∞ (X°). A similar result for the two-variable case is also proved.     

PROJECTIONS ON OPERATOR DUALS

Abstract

We discuss the existence of a projection with kernel K_b(E,F) ¹ (the annihilator of the quasi-compact operators) on the dual space of the space L _b,(E, F) of continous linear operators. Our results are proved in the context of Hausdorff locally convex spaces, but also provide extensions of recent results in the context of Banach spaces.     

AN EXTREME POINT WHICH IS ORTHOGONAL TO THE RANGE OF THE OPERATOR,CONJUGATED TO AN OPERATOR ON l 1

Abstract

In this note we prove that if a linear bounded operator T from l ₁ to a Banach space Y is not an isomorphism, then there exists an element f = (f ₁, f ₂,…) ε l _∞ such that |f _n| = 1 for every n and dist (f, T*Y*) = 1. This result we apply to Sidon sets in the theory of Fourier series.