Q-functions of Hermitian Contractions of Krein-Ovcharenko Type

In this paper operator-valued Q-functions of Krein-Ovcharenko type are introduced. Such functions arise from the extension theory of Hermitian contractive operators A in a Hilbert space ℌ. The definition is related to the investigations of M.G. Krein and I.E. Ovcharenko of the so-called Q_μ- and Q_M-functions. It turns out that their characterizations of such functions hold true only in the matrix valued case. The present paper extends the corresponding properties for wider classes of selfadjoint contractive extensions of A. For this purpose some peculiar but fundamental properties on the behaviour of operator ranges of positive operators will be used. Also proper characterizations for Q_μ- and Q_M-functions in the general operator-valued case are given. Shorted operators and parallel sums of positive operators will be needed to give a geometric understanding of the function-theoretic properties of the corresponding Q-functions.     

Inequalities for the Hadamard Weighted Geometric Mean of Positive Kernel Operators on Banach Function Spaces

Let K₁, . . . , K_n be positive kernel operators on a Banach function space. We prove that the Hadamard weighted geometric mean of K₁, . . . , K_n, the operator K, satisfies the following inequalities where || · ||and r(·) denote the operator norm and the spectral radius, respectively. In the case of completely atomic measure space we show some additional results. In particular, we prove an infinite-dimensional extension of the known characterization of those functions satisfying for all non-negative matrices A₁, . . . , A_n of the same order.     

On closures of joint similarity orbits

For an n-tuple T=(T₁,..., T_n) of operators on a Hilbert spacexxHx, the joint similarity orbit of T isxxSx(T)={VTV^–1 =(VT₁V^–1,...,VT_nV^–1): V is invertible onxxHx}. We study the structure of the norm closure ofxxSx, both in the case when T is commutative and when it is not. We first develop a Rota-model for the Taylor spectrum and use it to study n-tuples with totally disconnected Taylor spectrum, in particular quasinilpotent ones. We then consider limits of nilpotent n-tuples, and of normal n-tuples. For noncommuting n-tuples, we present a number of surprising facts relating the closure ofxxSx(T) to the Harte spectrum of T and the lack of commutativity of T. We show that a continuous function which is constant onxxSx(T) for all T must be constant. We conclude the paper with a detailed study of closed similarity orbits.Research partially supported by grants from the National Science Foundation.     

On inequalities for eigenvalues of 2 × 2 matrices with Schatten–von Neumann entries

Let SN_r (r ≥ 1) denote the Schatten-von Neumann ideal of compact operators in a separable Hilbert space. For the block matrix

the inequality

(p = 2; 3;?…?) is proved, where λ_k(A) (k = 1; 2;?…?) are the eigenvalues of A and N_r(.) is the norm in SN_r. Moreover, let P(z) = z²I + Bz + C (z ∈ ?) with B ∈ SN_2p, C ∈ SN_p. By z_k(P) (k = 1; 2;?…?) the characteristic values of the pencil P are denoted. It is shown that

In the case p = 1, sharper results are established. In addition, it is derived that

     

On the approximation of positive operators and the behaviour of the spectra of the approximants

LetT be a positive linear operator on the Banach latticeE and let (S _n ) be a sequence of bounded linear operators onE which converge strongly toT. Our main results are concerned with the question under which additional assumptions onS _n andT the peripheral spectra (S _n ) ofS _n converge to the peripheral spectrum (T) ofT. We are able to treat even the more general case of discretely convergent sequences of operators.     

Sequences in the range of a vector measure and Banach spaces satisfying Π1(X,l 1) = Π2(X,l 1)

If X is a Banach space such that Π₁(X,l ₁) = Π₂ (X, l ₁), we prove the following results: 1) A bounded sequence (x _n ) lies inside the range of some X-valued measure if and only if the operator (α _n ) ? l ₁ → Σ _n α _n x _n ? X is 1-summing, and 2) If A is a bounded subset of X lying in the range of some X ^??-valued measure, then A is necessarily contained in the range of some X-valued measure.     

Weyl’s Theorems for Some Classes of Operators

We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator T_μ of a group algebra L¹(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras.     

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.     

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.     

Quasi-similarp-hyponormal operators

IfA _i i=1, 2 are quasi-similarp-hyponormal operators such thatUi is unitary in the polar decompositionA _i =U _i |A _i |, then (A ₁)=(A ₂) and _c(A₁) = _e(A₂). Also a Putnam-Fuglede type commutativity theorem holds for p-hyponomral operators.     

Almost stable spectra and limits of similarities

Suppose that {D _n } is a sequence of invertible operators on a Hilbert space, andD _n T D _n ^–1 converges in norm toT ₀. Recently, H. Bercovici, C. Foias, and A. Tannenbaum have shown that if {D _n ^±1 n=1, 2,...} is contained in a finite dimensional subspace of operators, thenT andT ₀ must have the same spectral radius. Using this result, R. Teodorescu proved that the resolvents ofT andT ₀ have the same unbounded component. We show that in fact the spectra differ only by certain eigenvalues ofT ₀, and the spectrum ofT ₀ is obtained by filling in holes in the spectrum ofT; i.e., by adjoining (all, some, or none of the) bounded components of the resolvent ofT to the spectrum ofT.     

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)}     

Approximation of approximation numbers by truncation

LetA be a bounded linear operator onsome infinite-dimensional separable Hilbert spaceH and letA _n be the orthogonal compression ofA to the span of the firstn elements of an orthonormal basis ofH. We show that, for eachk1, the approximation numberss _k(A_n) converge to the corresponding approximation numbers _k(A) asn. This observation implies almost at once some well known results on the spectral approximation of bounded selfadjoint operators. For example, it allows us to identify the limits of all upper and lower eigenvalues ofA _n in the case whereA is selfadjoint. These limits give us all points of the spectrum of a selfadjoint operator which lie outside the convex hull of the essential spectrum. Moreover, it follows that the spectrum of a selfadjoint operatorA with a connected essential spectrum can be completely recovered from the eigenvalues ofA _n asn goes to infinity.     

Invariance of closed convex sets and domination criteria for semigroups

Let a and b be two positive continuous and closed sesquilinear forms on the Hilbert space H=L ²(, ). Denote by T=T(t) _t0and S=S(t) _t0the semigroups generated by a and b on H. We give criteria in terms of a and b guaranteeing that the semigroup T is dominated by S, i.e. |T(t)f|S(t)|f| for all t0 and fH. The method proposed uses ideas on invariance of closed convex sets of H under semigroups. Applications to elliptic operators and concrete examples are given.     

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.     

Natural Representations of the Multiplicity of an Analytic Operator-valued Function at an Isolated Point of the Spectrum

Representations are given for the multiplicity of an analytic operator-valued function A at an isolated point z₀ of the spectrum in the form of kernels and ranges of Hankel and Toeplitz matrices whose entries are derived from the Taylor coefficients of A and the Laurent coefficients of A⁻¹ about z₀. In two special cases the results can be expressed in terms of finite matrices: when A is a polynomial and when A⁻¹ has a pole at z₀. The latter case leads to the theory of Jordan chains.     

Reconstruction of a rational nonsquare matrix function from local data

In many problems the local zero-pole structure (i.e. locations of zeros and poles together with their orders) of a scalar rational functionw is a key piece of structure. Knowledge of the order of the pole or zero of the rational functionw at the point is equivalent to knowledge of the -module (where is the space of rational functions analytic at ). For the more intricate case of a rationalp×m matrix functionW, we consider the structure of the module as the appropriate analogue of zero-pole structure (location of zeros and poles together with directional information), where is the set of column vectors of heightm with entries equal to rational functions which are analytic at . Modules of the form in turn can be explicitly parametrized in terms of a collection of matrices (C ,A ,B ,B , ) together with a certain row-reduced(p–m)×m matrix polynomialP(z) (which is independent of ) which satisfy certain normalization and consistency conditions. We therefore define the collection (C ,A ,Z ,B , ,P(z)) to be the local spectral data set of the rational matrix functionW at . We discuss the direct problem of how to compute the local spectral data explicitly from a realizationW(z)=D+C(z–A) ^–1 B forW and solve the inverse problem of classifying which collections (C ,A ,Z ,B , ,P(z)) satisfying the local consistency and normalization conditions arise as the local spectral data sets of some rational matrix functionW. Earlier work in the literature handles the case whereW is square with nonzero determinant.     

On the Range of Elementary Operators

Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Let A = (A₁,A₂,.., A_n) and B = (B₁, B₂,.., B_n) be n-tuples in B(H), we define the elementary operator by In this paper we initiate the study of some properties of the range of such operators.     

A functional calculus for analytic generators ofC 0-groups

In [9] and [3] anF(S )-functional calculus for sectorial operators is constructed via the Dunford-Riesz integral. This calculus implicitely defines the well-known complex powers of such operators. Sectorial operators with bounded imaginary powers turn out to be of particular interest due to the remarkable Dore-Venni theorem. In [12] this theorem is proved via the theory of analytic generators ofC ₀-groups. These results suggest the existence ofF(S )-functional, calculi forC ₀-groups and their analytic generators. In this paper we show that such functional calculi indeed exsist, however the approach via the Dunford-Riesz integral is no longer viable. Therefore a different approach via an approximation argument is introduced. Existence and uniqueness theorems are given and we show how the functional calculi relate to known results. Examples illustrate the theory.