Self-adjoint Analytic Operator Functions: Local Spectral Function and Inner Linearization

In this note we continue the study of spectral properties of a self-adjoint analytic operator function A(z) that was started in [5]. It is shown that if A(z) satisfies the Virozub–Matsaev condition on some interval Δ₀ and is boundedly invertible in the endpoints of Δ₀, then the ‘embedding’ of the original Hilbert space into the Hilbert space , where the linearization of A(z) acts, is in fact an isomorphism between a subspace of and . As a consequence, properties of the local spectral function of A(z) on Δ₀ and a so-called inner linearization of the operator function A(z) in the subspace are established.      

Generalization of the determinants for trace-potent linear operators

SupposeA is a bounded linear operator on a separable Hilbert space withA ^m of trace class for some positive integerm. A generalized determinant for the operatorI–A is defined, its properties studied and this determinant is then used to exhibit an inversion formula forI–A.     

On perturbations of linear m-accretive operators on reflexive Banach spaces

In this note we prove some results on the m-accretivity of sums and products of linear operators. In particular we obtain the following theorem: LetA, B be two m-accretive operators on a reflexive Banach space. IfA is invertible and (A)^–1 B is accretive thenBA ^–1 andA+B are m-accretive.     

Convergence of an operator series

The paper gives a necessary and sufficient condition on the spectrum of a bounded linear operator on Banach space for the convergence of the series ₀ T(I-T ²) ⁿ . Some properties of the sum are investigated.     

Samuel multiplicity and Fredholm theory

In this note we combine methods from commutative algebra and complex analytic geometry to calculate the generic values of the cohomology dimensions of a commuting multioperator on its Fredholm domain. More precisely, we prove that, for a given Fredholm tuple T = (T ₁, ..., T _n ) of commuting bounded operators on a complex Banach space X, the limits exist and calculate the generic dimension of the cohomology groups H ^p (z − T, X) of the Koszul complex of T near z = 0. To deduce this result we show that the above limits coincide with the Samuel multiplicities of the stalks of the cohomology sheaves of the associated complex of analytic sheaves at z = 0.     

An eigenvalue problem for the numerical range of a bounded linear operator

LetX be a complex Lebesgue space with a unique duality mapJ fromX toX ^*, the conjugate space ofX. LetA be a bounded linear operator onX. In this paper we obtain a non-linear eigenvalue problem for (A)=sup{Re: W(A} whereW(A)={J(x)A(x)) : x=1}, under the assumption that (A) and the convex hull ofW(A) for some linear operatorsA onl _p , 2<p<.     

Subsolutions for abstract evolution equations

We study different notions of subsolutions for an abstract evolution equation du/dt+Auf where A is an m-accretive nonlinear operation in an ordered Banach space X with order-preserving resolvents. A first notion is related to the operator d/dt+A in the ordered Banach space L ¹(0, T; X); a second one uses the evolution equation du/dt+A uf where A :x{y;zy for some zAx}; other notions are also considered.     

Multiplication operators on Hilbert spaces of analytic functions

Let be a Hilbert space of functions analytic on a plane domain such that for every in the functional of evaluation at is bounded. Assume further that contains the constants and admits multiplication by the independent variable z, M_z, as a bounded operator. We give sufficient conditions for M_z to be reflexive.Received: 17 February 2004     

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.     

Some generalizations of the classical moment problem

The article is devoted to two generalizations of the classical power moment problem, namely: 1) instead of representing the moment sequence by ⁿ , a representation by polynomialsP _n (), ¹, connected with a Jacobi matrix, appears; 2) in the representation, instead of ⁿ , the expression ⁿ figures, where is a real generalized function (i.e., we investigate some infinite-dimensional moment problem).The work is partially supported by the DFG, Project 436 UKR 113/39/0 and by the CRDF, Project UM1-2090.     

Eigenfunctions of the hyperbolic composition operator

Letu inH ² be zero at one of the fixed points of a hyperbolic Möbius transform of the unit diskD. We will show, under some additional conditions onu, that the doubly cyclic subspaceS _u =V _n=– C ⁿ u contains nonconstant eigenfunctions of the composition operatorC . This implies that the cyclic subspace generated byu is not minimal. If there is an infinite dimensional minimal invariant subspace ofC (which is equivalent to the existance of an operator with only trivial invariant subspaces), then it is generated by a function with singularities at the fixed points of .     

Grothendieck’s comparison theorem and multivariable Fredholm theory

We use a variant of Grothendieck’s comparison theorem to show that, for a Fredholm tuple T ∈ L(X)ⁿ on a complex Banach space, there are isomorphisms . We conclude that a Fredholm tuple T ∈ L(X)ⁿ satisfies Bishop’s property (β) at z = 0 if and only if the vanishing conditions hold for . We apply these observations and results from commutative algebra to show that a graded tuple on a Hilbert space is Fredholm if and only if it satisfies Bishop’s property (β) at z = 0 and that, in this case, its cohomology groups can grow at most like k^p. Received: 14 January 2009     

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r _j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded, then there exists a nontrivial common invariant subspace for T ^* and (T − λ _j I)^*-1.     

On invariant subspaces of Volterra-type operators

The lattice of all the closed, invariant subspaces of the Volterra integration operator onL ²[0, 1] is equal to {B(a):a[0, 1]}, whereB(a)={fL ²[0, 1]:f=0 a.e. on [0,a]}. In order to extend this result to Banach function spaces we study the Volterra-type operatorV that was introduced in [7] for the case ofL ^p -spaces. Our main result characterizesL-closed subspaces of a Banach function spaceL that are invariant underV, whereL denotes the associate space ofL. In particular, if the norm ofL is order continuous and ifV is injective, then all the closed, invariant subspaces ofV are determined.This work was supported by the Research Ministry of Slovenia.     

Toeplitz operators with noncommuting symbols

LetM be a von Neumann algebra with a faithful normal tracial state and letH be a finite maximal subdiagonal subalgebra ofM. LetH ² be the closure ofH in the noncommutative Lebesgue spaceL ²(M). We consider Toeplitz operators onH ² whose symbol belong toM, and find that they possess several of the properties of Toeplitz operators onH ²( ) with symbol fromL ( ), including norm estimates, a Hartman-Wintner spectral inclusion theorem, and a characterisation of the weak^* continuous linear functionals on the space of Toeplitz operators.     

An algorithm for the linear complementarity problem with upper and lower bounds

In this paper, we adapt the octahedral simplicial algorithm for solving systems of nonlinear equations to solve the linear complementarity problem with upper and lower bounds. The proposed algorithm generates a piecewise linear path from an arbitrarily chosen pointz ⁰ to a solution point. This path is followed by linear programming pivot steps in a system ofn linear equations, wheren is the size of the problem. The starting pointz ⁰ is left in the direction of one of the 2 ⁿ vertices of the feasible region. The ray along whichz ⁰ is left depends on the sign pattern of the function value atz ⁰. The sign pattern of the linear function and the location of the points in comparison withz ⁰ completely govern the path of the algorithm.This research is part of the VF-Program Equilibrium and Disequilibrium in Demand and Supply, approved by the Netherlands Ministry of Education, Den Haag, The Netherlands.     

Direct and inverse spectral problems for generalized strings

Let the functionQ be holomorphic in he upper half plane ⁺ and such that ImQ(z 0 and ImzQ(z) 0 ifz ⁺. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf + fdm = 0; f(0–) = 0. In this note we show that the set of functionsQ which are holomorphic in ⁺ and such that the kernel has negative squares of ⁺ and ImzQ(z) 0 ifz ⁺ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf +f dm + ² fdD = 0 on [0,L),f(0–) = 0. Here is the number of pointsx whereD increases or 0 >m(x + 0) –m(x – 0) –; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.To the memory of M.G. Krein with deep gratitude and affection.This author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 09832     

Sesquitransitive and Localizing Operator Algebras

An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x _n ) in B there exists a subsequence and a bounded sequence (A _k ) in the algebra such that converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every there exists A in the algebra such that and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive rings. The second and the third authors were supported by NSERC.