Small perturbations of canonical systems

We consider a singular two-dimensional canonical systemJy=–zHy on [0, ) such that at Weyl's limit point case holds. HereH is a measurable, real and nonnegative definite matrix function, called Hamiltonian. From results of L. de Branges it follows that the correspondence between canonical systems and their Titchmarsh-Weyl coefficients is a bijection between the class of all Hamiltonians with trH=1 and the class of Nevanlinna functions. In this note we show how the HamiltonianH of a canonical system changes if its Titchmarsh-Weyl coefficient or the corresponding spectral measure undergoes certain small perturbations. This generalizes results of H. Dym and N. Kravitsky for so-called vibrating strings, in particular a generalization of a construction principle of I.M. Gelfand and B.M. Levitan can be shown.     

Subordinate solutions and spectral measures of canonical systems

The theory of 2×2 trace-normed canonical systems of differential equations on ?⁺ can be put in the framework of abstract extension theory, cf. [9]. This includes the theory of strings as developed by I.S. Kac and M.G. Kre?n. In the present paper the spectral properties of such canonical systems are characterized by means of subordinate solutions. The theory of subordinacy for Schrödinger operators on the halfline ?⁺, was originally developed D.J. Gilbert and D.B. Pearson. Its extension to the framework of canonical systems makes it possible to describe the spectral measure of any Nevanlinna function in terms of subordinate solutions of the corresponding trace-normed canonical system, which is uniquely determined by a result of L. de Branges.     

Two-Dimensional Trace-Normed Canonical Systems of Differential Equations and Selfadjoint Interface Conditions

The class of two-dimensional trace-normed canonical systems of differential equations on is considered with selfadjoint interface conditions at 0. If one or both of the intervals around 0 are H-indivisible the interface conditions which give rise to selfadjoint relations (multi-valued operators) are determined. It is shown that the corresponding generalized Fourier transforms are partially isometric. Compression to the halfline (0, ) results in boundary conditions which depend on the eigenvalue parameter involving a fractional linear transform of the Titchmarsh-Weyl coefficient of the halfline (–, 0). The corresponding generalized Fourier transforms are isometric except possibly on a one-dimensional subspace where they are contractive.     

Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical systemJy=–Hy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The formal defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ) to a trace-normed system on which is in the limit-point case at ±. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.     

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     

On some operator monotone functions

We try to find a continuous functionu defined on a real right half-line with the range (0, ) such thatu ^–1 is operator monotone. We then look for another functionv such thatv(u ^–1) is operator monotone, namely,u(A)u(B) impliesv(A)v(B) for self-adjoint operatorsA andB.     

On the peripheral spectrum of monic operator polynomials with positive coefficients

I wish to thank R. Nagel for his guidance and suggestions in the preparation of this paper. Also, I would like to thank G. Greiner and F. Räbiger for many interesting and helpful discussions.     

Multi-variable weighted geometric means of positive definite matrices

We define a family of weighted geometric means {G(t;ω;A)}_t∈[0,1]ⁿ where ω and A vary over all positive probability vectors in Rⁿ and n-tuples of positive definite matrices resp. Each of these weighted geometric means interpolates between the weighted ALM (t=0_n) and BMP (t=1_n) geometric means (ALM and BMP geometric means have been defined by Ando-Li-Mathias and Bini-Meini-Poloni, respectively.) We show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean.     

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.     

Two-sided estimates on singular values for a class of integral operators on the semi-axis

(Quasi)-norms inC _p andC _{p, w} of weighted operators of the integration of (fractional) order are estimated. It is shown that, in most cases, the estimates obtained are sharp both in order and in function classes for the weight function involved.     

Nonselfadjoint spectral problems for linear pencilsN-λP of ordinary differential operators with λ-linear boundary conditions: Completeness results

We study nonselfadjoint spectral problems for ordinary differential equationsN(y)–P(y)=0 with -linear boundary conditions where the orderp of the differential operatorP is less than the ordern ofN. The present paper addresses the question of the completeness of the eigenfunctions and associated functions in the Sobolev spacesW ₂ ^k (0,1) fork=0,1,...,n. To this end we associate a pencil – of operators acting fromL ₂(0,1) to the larger spaceL ₂x(0,1) ⁿ with the given problem. We establish completeness results for normal problems in certain finite codimensional subspaces ofW ₂ ^k (0,1) which are characterized by means of Jordan chains in 0 of the adjoint of the compact operator = ^–1.Dedicated to Professor Heinz Langer on the occasion of his 60th birthday     

Completion of operator partial matrices associated with chordal graphs

H. Dym and I. Gohberg established in [6] necessary and sufficient conditions for the existence and uniqueness of an invertible block matrix F=(F_ij)_i,j=1,...,n such that F_ij=R_ij for |i–j|m and F^–1 has a band triangular factorization and so (F^–1)_ij=0 for |i–j|>m. Here R_ij, |i–j|m are given block matrices.The aim of this paper is to generalize these results in two directions. First, we shall allow R_ij to be an (linear bounded) operator acting between the Hilbert spaces H_j and H_i. Secondly, the set of indices of the given R_ij will be more general than banded ones. In fact, we will consider index sets which have an associated graph which is chordal. The case of partial positive operator matrices is also discussed.     

Algebraic reflexivity of some subsets of the isometry group

Let X be a compact first countable space. In this paper we show that the set of isometries of C(X) that are involutions is algebraically reflexive. As a consequence of a recent work of Botelho and Jamison this leads to the conclusion that the set of generalized bi-circular projections on C(X) is also algebraically reflexive. We also consider these questions for the space C(X,E) where E is a uniformly convex Banach space.     

Dissipative q-Dirac operator with general boundary conditions

In this paper, we consider the symmetric q-Dirac operator. We describe dissipative, accumulative, self-adjoint and the other extensions of such operators with general boundary conditions. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete.     

Norms of the singular integral operator with Cauchy Kernel along certain contours

The norm of the above-mentioned operatorS is computed on the unions of parallel lines or concentric circles. The upper bound is found for its norm on the ellipse. In case of weighted spaces on the unit circle, the exact norm is found for some rational weights, and necessary and sufficient conditions on the weight are established, under which the essential norm ofS equals 1.     

Spectral properties of cosine operator functions

LetA be the generator of a cosine functionC _t ,t R in a Banach spaceX; we shall connect the existence and uniqueness of aT-periodic mild solution of the equationu = Au + f with the spectral property 1 (C _T ) and, in caseX is a Hilbert space, also with spectral properties ofA. This research was supported in part by DAAD, West Germany.