Finite rank perturbations of contractions

We study finite rank perturbations of contractions of classC _.0 with finite defect indices. The completely nonunitary part of such a perturbation is also of classC _.0, while the unitary part is singular. When the defect indices of the original contraction are not equal, it can be shown that almost always (with respect to a suitable measure) the perturbation has no unitary part.     

Dilation to the unilateral shifts

The classical result of Foias says that an operator power dilates to a unilateral shift if and only if it is aC ₀ contraction. In this paper, we consider the corresponding question of dilating to a unilateral shift. We show tht for contractions with at least one defect index finite, dilation and power dilation to some unilateral shift amount to the same thing. The only difference is on the minimum multiplicity of the unilateral shift to which the contraction can be (power) dilated. We also obtain a characterization of contractions which are finite-rank perturbations of a unilateral shift, generalizing the rank-one perturbation result of Nakamura.     

Polar Decompositions of C 0(N) Contractions

Let A be a bounded linear operator on a complex separable Hilbert space H. We show that A is a C₀(N) contraction if and only if , where U is a singular unitary operator with multiplicity and x₁, . . . , x_d are orthonormal vectors satisfying . For a C₀(N) contraction, this gives a complete characterization of its polar decompositions with unitary factors.     

Integral Equations and Operator Theory

A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C₀ contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of Aⁿ, n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A^* has trivial kernel.     

Some results on b-AM-compact operators

We show that for positive operator B : E → E on Banach lattices, if there exists a positive operator S : E → E such that:1.SB ≤ BS;2.S is quasinilpotent at some x₀ > 0; 3.S dominates a non-zero b-AM-compact operator, then B has a non-trivial closed invariant subspace. Also, we prove that for two commuting non-zero positive operators on Banach lattices, if one of them is quasinilpotent at a non-zero positive vector and the other dominates a non-zero b-AM-compact operator, then both of them have a common non-trivial closed invariant ideal. Then we introduce the class of b-AM-compact-friendly operators and show that a non-zero positive b-AM- compact-friendly operator which is quasinilpotent at some x₀ > 0 has a non-trivial closed invariant ideal.     

Domination of uniformly continuous semigroups

We prove that a bounded operator on a Banach lattice, satisfying a growth condition, is regular. Also, we prove that the generator of a C ₀-semigroup on such a lattice for which such an operator exists is bounded.     

On the Existence of Solutions to the Operator Riccati Equation and the tan Θ Theorem

Let A and C be self-adjoint operators such that the spectrum of A lies in a gap of the spectrum of C and let d > 0 be the distance between the spectra of A and C. Under these assumptions we prove that the best possible value of the constant c in the condition guaranteeing the existence of a (bounded) solution to the operator Riccati equation XA–CX+XBX = B* is equal to We also prove an extension of the Davis-Kahan tan theorem and provide a sharp estimate for the norm of the solution to the Riccati equation. If C is bounded, we prove, in addition, that the solution X is a strict contraction if B satisfies the condition and that this condition is optimal.     

Fredholm composition operators on spaces of holomorphic functions

Composition operators on vector spaces of holomorphic functions are considered. Necessary conditions that range of the operator is of a finite codimension are given. As a corollary of the result it is shown that a composition operatorC on a certain Banach space of holomorphic functions on a strictly pseudoconvex domain withC ² boundary or a polydisc or a compact bordered Riemann surface or a bounded domainD such that intD = D is invertible if and only if it is a Fredholm operator if and only if is a holomorphic automorphism.     

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.     

Operator Matrices: SVEP and Weyl’s Theorem

It is known that if and are Banach space operators with the single-valued extension property, SVEP, then the matrix operator has SVEP for every operator and hence obeys Browder’s theorem. This paper considers conditions on operators A, B, and M₀ ensuring Weyls theorem for operators M_C.     

On the norm of singular integral operator on curves with cusps

Applying the results on singular integral operators with the complex conjugation on curves with cusps (see R. Duduchava, T. Latsabidze, A. Saginashvili, 1992, 1994) the explicit formula for the local norm of the Cauchy singular integral operator on a curve with cusps in a Lebesgue space with an exponential weightL ₂ (, ) is obtained. For curves with angles the formula was already known (see R. Avedanio, N. Krupnik, 1988).     

Comparing different classes of absolutely summing multilinear operators

Let C denote the composition operator defined on the standard Hardy spaces H^p as where is an analytic self-map of the unit disk in the complex plane. In this paper we discuss those invariant subspaces of C in H^p which are invariant under the shift operator, We restrict our attention to the case where is an inner function. Our main result characterises these invariant subspaces. We also consider C when restricted to such an invariant subspace and we describe the structure of the operator and find a formula for the essential spectral radius.Received: 27 January 2004     

Joint similarity problems and the generation of operator algebras with bounded length

There is a pair of commuting operators (T ₁,T ₂) on Hilbert space such that eachT ₁ andT ₂ is similar to a contraction but the pair (T ₁,T ₂) is not similar to a pair of contractions. There is a pair of commuting unitarizable representations (₁,₂) on the free group withN2 generators such that (₁,₂) is not similar to a pair of unitary representations. In connection with these examples, we introduce and study a notion of length for aC ^*-algebra (or an operator algebra) generated by two subalgebras, which is analogous to the minimum length of a word in the generators of a group.Partially supported by the N.S.F.     

The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (non-self-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C^?-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the wot-closures of these algebras as well.     

Invariant subspaces and localizable spectrum

LetT L(X) be a continuous linear operator on a complex Banach spaceX. We show thatT possesses non-trivial closed invariant subspaces if its localizable spectrum _loc(T) is thick in the sense of the Scott Brown theory. Since for quotients of decomposable operators the spectrum and the localizable spectrum coincide, it follows that each quasiaffine transformation of a Banach-space operator with Bishop's property () and thick spectrum has a non-trivial invariant subspace. In particular it follows that invariant-subspace results previously known for restrictions and quotients of decomposable operators are preserved under quasisimilarity.     

Aluthge Transforms of Complex Symmetric Operators

If denotes the polar decomposition of a bounded linear operator T, then the Aluthge transform of T is defined to be the operator . In this note we study the relationship between the Aluthge transform and the class of complex symmetric operators (T iscomplex symmetric if there exists a conjugate-linear, isometric involution so that T = CT*C). In this note we prove that: (1) the Aluthge transform of a complex symmetric operator is complex symmetric, (2) if T is complex symmetric, then and are unitarily equivalent, (3) if T is complex symmetric, then if and only if T is normal, (4) if and only if T ² = 0, and (5) every operator which satisfies T ² = 0 is necessarily complex symmetric. This work partially supported by National Science Foundation Grant DMS 0638789.     

Scattering Matrix for Magnetic Potentials with Coulomb Decay at Infinity

We consider the Schrödinger operator H in the space $ L_{2}(\mathbb{R}^{d})$ with a magnetic potential A(x) decaying as $ \vert x\vert^{-1} $ at infinity and satisfying the transversal gauge condition <A(x), x > = 0. Our goal is to study properties of the scattering matrix S() associated to the operator H. In particular, we find the essential spectrum _ess of S() in terms of the behaviour of A(x) at infinity. It turns out that _ess(S()) is normally a rich subset of the unit circle $\mathbb{T}$ or even coincides with $\mathbb{T}$. We find also the diagonal singularity of the scattering amplitude (of the kernel of S() regarded as an integral operator). In general, the singular part S₀ of the scattering matrix is a sum of a multiplication operator and of a singular integral operator. However, if the magnetic field decreases faster than $ \vert x\vert^{-1} $ for d 3 (and the total magnetic flux is an integer times 2 for dd = 2), then this singular integral operator disappears. In this case the scattering amplitude has only a weak singularity (the diagonal Dirac function is neglected) in the forward direction and hence scattering is essentially of short-range nature. Moreover, we show that, under such assumptions, the absolutely continuous parts of the operators S() and S₀ are unitarily equivalent. An important point of our approach is that we consider S() as a pseudodifferential operator on the unit sphere and find an explicit expression of its principal symbol in terms of A(x). Another ingredient is an extensive use (for d 3) of a special gauge adapted to a magnetic potential A(x).     

Perturbation of the Drazin inverse for closed linear operators

We investigate the perturbation of the Drazin inverse of a closed linear operator recently introduced by second author and Tran, and derive explicit bounds for the perturbations under certain restrictions on the perturbing operators. We give applications to the solution of perturbed linear equations, to the asymptotic behaviour ofC ₀-semigroups of linear operators, and to perturbed differential equations. As a special case of our results we recover recent perturbation theorems of Wei and Wang.     

Hypercyclic composition operators on Hv0‐spaces

We consider analytic self‐maps φ on $\mathbf {D}$ and prove that the composition operator C_φ acting on $H_{v}^0$ is hypercyclic if φ is an automorphism or a hyperbolic non‐automorphic symbol with no fixed point. We give examples of weights v and parabolic non‐automorphisms φ on $\mathbf {D}$ which yield non‐hypercyclic composition operators C_φ on $H_{v}^0$.