Coherent states,inverse mapping formula for Weyl transformations and application to equations of KdV type

In this paper, we consider the trace of generalized operators and inverse Weyl transformation. First of all we repeat the definition of test operators and generalized operators given in [18], denotingL ²(R) byH.     

SOME FUNCTIONAL ANALYTIC PROPERTIES OF COMPOSITION OPERATORS

Abstract

This is a report on a number of recent results on composition operators which map, for 0 < p ? q ∞, the Hardy space H^p (on the unit disk in the complex plane) into H ^q. Attention is focused on questions of boundedness (existence), compactness, order boundedness and, in connection with the latter, on relating the absolutely summing and nuclearity character as well as special factorization properties of the operator to function theoretic properties of the defining symbol. Moreover, tools are provided to show that certain classes of operators can well be distinguished already on the level of composition operators.     

Maps preserving the harmonic mean or the parallel sum of positive operators

Let H be a complex Hilbert space. The symbol A!B stands for the harmonic mean of the positive bounded linear operators A,B on H in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to that operation. We prove that any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on H. Similar results concerning the parallel sum and the arithmetic mean in the place of the harmonic mean are also presented.     

Unbounded Toeplitz Operators

This partly expository article develops the basic theory of unbounded Toeplitz operators on the Hardy space H ², with emphasis on operators whose symbols are not square integrable. Unbounded truncated Toeplitz operators on coinvariant subspaces of H ² are also studied. In memory of Paul R. Halmos     

Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H ¹ and give a new proof of the old result stating that the Hankel operator H _a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H _a to be compact on H ¹. The Fredholm properties of Toeplitz operators on H ¹ are studied for symbols in a Banach algebra similar to C + H ^∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H ¹ and H ². The first author was partially supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) and by the Greek Research Program “Pythagoras 2” (75% European funds and 25 National funds). The second author was fully supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) while he visited the first author at the University of Crete and later by the Academy of Finland Project 207048.     

Geometric interpolation in p-Schatten class

The aim of this work is to apply the complex interpolation method to norms of n-tuples of operators in the p-Schatten classes of a Hilbert space H. The norms considered define Finsler metrics in a certain manifold of positive operators, and can be regarded as weighted p-norms, the weight being a positive invertible operator. As a by-product, we obtain Clarkson's type inequalities.     

On the Approximation of Zeros of Non-Self Monotone Operators

In this article, we study the approximation of common zeros of non-self inverse strongly monotone operators defined on a closed convex subset C of a Hilbert space H. For a non-self family of operators, we introduce an iterative algorithm without relying on projections. Approximation of common fixed points for finite families of non-self strict pseudo-contractions in the sense of Browder-Petryshyn is also obtained. The novelty of our algorithm is that the coefficients are not given a priori and no assumptions are made on them, but they are constructed step by step in a natural way.     

On explicit inversion of a subclass of operators with <Emphasis Type="Italic">D</Emphasis>-difference kernels and Weyl theory of the corresponding canonical systems

Explicit inversion formulas for a subclass of integral operators with D-difference kernels on a finite interval are obtained. A case of the positive operators is treated in greater detail. An application to the inverse problem to recover canonical system from a Weyl function is given.     

Factorization of staircase class operators in Hilbert space

A class of linear bounded staircase operators (H, G spaces) defined by (1) with two infinite sequences of orthogonal decompositions ofH and chain property (2) is considered. Necessary and sufficient conditions for the factorizationZ=XY are obtained, whereX, Y are block-diagonal, bounded, andY has a bounded inverse. All the pairs (X, Y) are explicitly constructed. These conditions are specialized for finite and infinite dimensions of the blocks ofX, Y and for differentX, Y. A direct application to bitriangular and biquasitriangular operators is indicated.     

A Generalization of the Direct and Inverse Problem for the Radial Schrödinger Equation

We consider the operators H₀= ?d²/dr² and H₁ = ?d²/dr² + V(r) (0< r< ∞) acting on a Hilbert space of complex functions f(r) such that the subspaces in which the operators are defined consist of twice differentiable functions which satisfy the boundary condition (d/dr)f(0) = αf(0). H₁ and H₀ are Hermitian in this subspace. Assuming V(r)→0 as r→∞ sufficiently rapidly, the scattering operator formalism is set up for the direct scattering problem. Next we consider the inverse problem of determining V(r) from H₀ and the spectral measure function for the spectrum of H₁ through the use of an appropriate Gelfand-Levitan equation. It is shown that generally the value of α associated with H₁ differs from that for H₀, i.e., H₁ and H₀ generally operate in different subspaces. Thus scattering cannot be defined. However, by changing the spectral measure function, one obtains a new Gelfand-Levitan equation such that H₁ is the same as before [i.e., α and V(r) are the same] from the operator H₀, which uses the same value of α as H₁. Thus H₁ and the new H₀ operate in the same subspace of Hilbert space, and scattering can be defined. The process of obtaining the new H₀ after finding H₁ from the old H₀ is somewhat analogous to renormalization in field theory, where a new H₀ is picked to have properties compatible with H₁. A necessary and sufficient condition on the spectral data is given which makes the domains of H₀ and H₁ coincide and thus makes “renormalization” unnecessary. The direct problem is a generalization of the usual l=0 radial Schrödinger equation. The inverse problem is a generalization of the corresponding inverse problem. It is also a generalization of the case α=0 for H₀ considered by Gelfand and Levitan in their early work on the inverse spectral problem. An incompletely understood connection of the inverse problem for the radial equation to solutions of the Korteweg-deVries equation in the half space is discussed. The existence of such a connection is one of the motivations for studying the generalized radial Schrodinger equation.     

Kolmogorov equations for measures

We consider a semigroup of operators in the Banach space C _b (H) of uniformly continuous and bounded functions on a separable Hilbert space H. We prove an existence and uniqueness result for a measure valued equation involving this class of semigroups. Then we apply the result to the transition semigroup and the Kolmogorov operator corresponding to a stochastic PDE in H. For this purpose, we characterize the generator of the transition semigroup on a core.      

Direct and Inverse Spectral Theory of One-dimensional Schrödinger Operators with Measures

We present a direct and rather elementary method for defining and analyzing one-dimensional Schrödinger operators H = −d²/dx² + μ with measures as potentials. The basic idea is to let the (suitably interpreted) equation −f′′ + μ f = zf take center stage.We show that the basic results from direct and inverse spectral theory then carry over to Schrödinger operators with measures.     

Factorization along commutative subspace lattices

A positive invertible operatorT is said to be factorable along a commutative subspace latticeL if there is an invertible operatorA inAlg L whose inverse is also inAlg L and such thatT=A*A. We investigate a number of conditions that are equivalent to factorability of a given operator along a latticeL. As a byproduct, we derive a condition that guarantees that the latticeT L, defined as {range(TE) E L} is commutative. Applications are suggested to the particular case of factoringL functions via analytic Toeplitz operators on the polydisc.     

On Some Product of Two Unbounded Self-Adjoint Operators

We give a spectral analysis of some unbounded normal product HK of two self-adjoint operators H and K (which appeared in [7]) and we say why it is not self-adjoint even if the spectrum of one of the operators is sufficiently “asymmetric”. Then, we investigate the self-adjointness of KH (given it is normal) for arbitrary self-adjoint H and K by giving a counterexample and some positive results and hence finishing off with the whole question of normal products of self-adjoint operators (appearing in [1, 7, 12]). The author was supported in part by CNEPRU: B01820070020 (Ministry of Higher Education, Algeria).     

<Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>-calculus for Products of Non-Commuting operators

It is shown that the product of two sectorial operators A and B admits a bounded H^∞-calculus on a Banach space X provided suitable commutator estimates and Kalton-Weis type assumptions on A and B are satisfied.     

Schur spaces and weighted spaces of type H ∞

Abstract

We extend some results related to composition operators on H _υ(G) to arbitrary linear operators on H _υ0(G) and H _υ(G). We also give examples of rank-one operators on H _υ(G) which cannot be approximated by composition operators.     

L p-Theory of elliptic differential operators on manifolds of bounded geometry

This paper is devoted to some of the properties of uniformly elliptic differential operators with bounded coefficients on manifolds of bounded geometry in L ^pspaces. We prove the coincidence of minimal and maximal extensions of an operator of a considered type with a positive principal symbol, the existence of holomorphic semigroup, generated by it, and the estimates of L ^p-norms of the operators of this semigroup. Some spectral properties of such operators in L ^pspaces are also studied.     

Approximators to generalized inverses as regularizers of ill-posed problems

Let H₁ and H₂ be Hilbert spaces and let B be a closed linear operator mapping a dense subset of H₁ into H₂. Several families of approximators which converge t o the orthogonal or Moore-Penrose generalized inverse B^? are constructed. Additionally, the approximators are shown t o provide regularization operators for the equation Bx=y. Some of the results are extended to dissipative operators on reflexive and general Banach spaces.     

THE UNIQUENESS OF OPERATIONAL QUANTITIES IN VON NEUMANN ALGEBRAS

Abstract

Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N].     

Some inequalities involving quadratic forms of operator means

Let T and S be two self-adjoint positive invertible operators of a complex Hilbert space H. In this paper, we investigate some inequalities involving the quadratic forms of the weighted arithmetic and harmonic means of T and S. Application for operator entropies is also discussed.