Positive Decompositions of Selfadjoint Operators

Given a linear bounded selfadjoint operator a on a complex separable Hilbert space ${\mathcal{H}}Given a linear bounded selfadjoint operator a on a complex separable Hilbert space H{\mathcal{H}}, we study the decompositions of a as a difference of two positive operators whose ranges satisfy an angle condition. These decompositions are related to the canonical decompositions of the indefinite metric space (H,á ,  ?_a){(\mathcal{H},\langle\,, \,\rangle_a)}, associated to a. As an application, we characterize the orbit of congruence of a in terms of its positive decompositions.     

Dissipative Sturm–Liouville Operators in Limit-Point Case

Dissipative singular Sturm–Liouville operators are studied in the Hilbert space L_w²[a,b) (–<a<b), that the extensions of a minimal symmetric operator in Weyls limit-point case. We construct a selfadjoint dilation of the dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function in terms of the Titchmarsh–Weyl function of a selfadjoint operator. Finally, in the case when the Titchmarsh–Weyl function of the selfadjoint operator is a meromorphic in complex plane, we prove theorems on completeness of the system of eigenfunctions and associated functions of the dissipative Sturm–Liouville operators. Mathematics Subject Classifications (2000) 47A20, 47A40, 47A45, 34B20, 34B44, 34L10.     

Compactness of Bochner representable operators on Orlicz spaces

We study compactness properties of linear operators from an Orlicz space L^Φ provided with a natural mixed topology to a Banach space (X, || · ||_X). We derive that every Bochner representable operator is -compact. In particular, it is shown that every Bochner representable operator is (τ(L^∞, L¹), || · ||_X)-compact.      

Selfadjoint operators in S-spaces

We study S-spaces and operators therein. An S-space is a Hilbert space with an additional inner product given by , where U is a unitary operator in . We investigate spectral properties of selfadjoint operators in S-spaces. We show that their spectrum is symmetric with respect to the real axis. As a main result we prove that for each selfadjoint operator A in an S-space we find an inner product which turns S into a Krein space and A into a selfadjoint operator therein. As a consequence we get a new simple condition for the existence of invariant subspaces of selfadjoint operators in Krein spaces, which provides a different insight into this well-know and in general unsolved problem.     

Uniform Schauder estimates for regularized hypoelliptic equations

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: in , where Δ is the Laplace operator, m < n, and the limit operator is hypoelliptic. Here we establish Schauder’s estimates, uniform with respect to the parameter ϵ, of solution of the approximated equation L _ϵ u = f, using a modification of the lifting technique of Rothschild and Stein. These estimates can be used in particular while studying regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.      

Wiener–Hopf Operators on Spaces of Functions on $${\mathbb{R}}^{+}$$ with Values in a Hilbert Space

A Wiener–Hopf operator on a Banach space of functions on is a bounded operator T such that P ⁺ S _−a TS _a = T, a ≥ 0, where S _a is the operator of translation by a. We obtain a representation theorem for the Wiener–Hopf operators on a large class of functions on with values in a separable Hilbert space.      

Quadratic transformations for a function that generalizes 2F1 and the Askey-Wilson polynomials

In previous papers we introduced and studied a ‘relativistic’ hypergeometric function R(a ₊, a ₋, c; v, ) that satisfies four hyperbolic difference equations of Askey-Wilson type. Specializing the family of couplings c∊ to suitable two-dimensional subfamilies, we obtain doubling identities that may be viewed as generalized quadratic transformations. Specifically, they give rise to a quadratic transformation for ₂ F ₁ in the ‘nonrelativistic’ limit, and they yield quadratic transformations for the Askey-Wilson polynomials when the variables v or are suitably discretized. For the general coupling case, we also study the bearing of several previous results on the Askey-Wilson polynomials. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D45, 39A70     

Binormal operator and *-Aluthge transformation

Let T = U｜T｜ be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation T = ｜T｜^1/2 U｜T｜^1/2 is called the Aluthge transformation and Tn means the n-th Aluthge transformation. Similarly, the transformation T（＊）=｜T＊｜^1/2 U｜T＊｜＆1/2 is called the ＊-Aluthge transformation and Tn^（＊） means the n-th ＊-Aluthge transformation. In this paper, firstly, we show that T（＊） = UV｜T^（＊）｜ is the polar decomposition of T（＊）, where ｜T｜^1/2 ｜T^＊｜^1/2 = V｜｜T｜^1/2 ｜T^＊｜^1/2｜ is the polar decomposition. Secondly, we show that T（＊） = U｜T^（＊）｜ if and only if T is binormal, i.e., [｜T｜, ｜T^＊｜]=0, where [A, B] = AB - BA for any operator A and B. Lastly, we show that Tn^（＊） is binormal for all non-negative integer n if and only if T is centered, and so on.     

An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures

We study hypersurfaces in Euclidean space whose position vector x satisfies the condition L _k x = Ax + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed , is a constant matrix and is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form , with . This extends a previous classification for hypersurfaces in satisfying , where is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].      

Increasing Monotone Operators in Banach Space

An operator mapping a separable reflexive Banach space X into the dual space X is called increasing if as . Necessary and sufficient conditions for the superposition operators to be increasing are obtained. The relationship between the increasing and coercive properties of monotone partial differential operators is studied. Additional conditions are imposed that imply the existence of a solution for the equation with an increasing operator A.     

On a spin conformal invariant on manifolds with boundary

Let M be an n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric g, a spin structure σ and a chirality operator Γ. We define and study some properties of a spin conformal invariant given by:

On <Emphasis Type="Italic">Γ</Emphasis>-convergence of quadratic functionals in the plane

In this paper, we prove that if a sequence of homeomorphisms , with bounded planar domains, of Sobolev space has uniformly equibounded distortions in EXP(Ω) and weakly converges to f in then the matrices A(x, f _j ) of the corresponding Laplace-Beltrami operators Γ-converge in the Orlicz–Sobolev space , where Q(t) = t ²log(e + t), to the matrix A(x, f) of the Laplace-Beltrami operator associated to f.      

Regular orbits and positive directions

Let A be a bounded linear operator defined on a separable Banach space X. Then A is said to be supercyclic if there exists a vector x ∈ X (later called supercyclic for A), such that the projective orbit is dense in X. On the other hand, A is said to be positive supercyclic if for each supercyclic vector x, the positive projective orbit, is dense in X. Sometimes supercyclicity and positive supercyclicity are equivalent. The study of this relationship was initiated in [14] by F. León and V. Müller. In this paper we study positive supercyclicity for operators A of the form , with , defined on . We will see that such a problem is related with the study of regular orbits. The notion of positive directions will be central throughout the paper.      

Nondissipative curves in Hilbert spaces having a limit of the corresponding correlation function

In this work a class of nondissipative curves in Hilbert spaces whose correlation functions have a limit ast± is presented. These curves correspond to a class of nondissipative basic operators that are a coupling of a dissipative operator and an antidissipative one. The wave operators and the scattering operator for the couple (A ^*, A) ( ) are obtained. The present work is a continuation and a generalization of the investigations of K.Kirchev and V.Zolotarev [1, 2, 3] on the model representations of curves in Hilbert spaces where the respective semigroup generator is a dissipative operator. This article includes four parts. A new form of the triangular model of M.S. Livic ([4, 5]) for the considered operators is introduced in the first part by the help of a suitable representation of the selfadjoint operatorL. This allows us to describe the studied class of nondissipative curves. The second part studies some results concerning the application of the analogue for multiplicative integrals of the well-known Privalov's theorem ([6]) about the limit values in the scalar case. This analogue is a reconstruction of measure by limit values in Stieltjes-Perron's style and it is obtained by L.A. Sakhnovich ([7]). Another problem, considered in the second part is the analogue inC ^m of the classical gamma-function and several properties for further consideration. In the third part the asymptotics of the studied curves corresponding to the nondissipative operators-couplings of a dissipative and an antidissipative operator with absolutely continuous real spectra and the limits of their correlation functions are obtained In the fourth part a scattering theory of a couple (A ^*, A) with a nondissipative operatorA from is constructed as in the selfadjoint case ([8, 9, 10]) and in the dissipative case ([7]). These results show an interesting new effect: the studied nondissipative case is near to the dissipative one.Partially supported by Grant MM-810/98 of MESC     

A Note on Toeplitz Operators in Schatten-Herz Classes Associated with Rearrangement-invariant Spaces

In recent papers, B. Choe, H. Koo, K. Na (see [3]) and Loaiza, M. Lopez-Garcia e S. Perez-Esteva (see [5]) studied conditions in order to a Toeplitz operator, acting on the harmonic Bergman space over the unit ball in and on analytic Bergman space on the unit disk in the complex plane, respectively, belong to the so-called Schaten-Herz class. The purpose of this note is to prove necessary and sufficient conditions in order to a Toeplitz operator T _μ with positive symbol, acting on the harmonic Bergman space on the unit ball in belong to a Schatten-Herz class S ^F _E , associated with a pair of rearrangement invariant sequence spaces E and F. The conditions involve the Berezin transform of its symbol and the average function. on some euclidian discs. The main point is the characterization of Toeplitz operators, that belong to Schatten ideals S _E associated with an arbitrary rearrangement invariant sequence space E.      

Bismut superconnections and the Chern character for Dirac operators on foliated manifolds

In this paper, we show how to define a Bismut superconnection for generalized Dirac operators defined along the leaves of a compact foliated manifoldM. Using the heat operator of the curvature of the superconnection, we define a (nonnormalized) Chern character for the Dirac operator, which lies in the Haefliger cohomology of the foliation. Rescaling the metric onM by 1/a and lettinga 0, we obtain the analog of the classical cohomological formula for the index of a family of Dirac operators. In certain special cases, we can also compute the limit asa and show that it is the Chern character of the index bundle given by the kernel of the Dirac operator. Finally, we discuss the relation of our results with the Chern character in cyclic cohomology.     

The Dolbeault complex with weights according to normal crossings

In the present paper, we define a Dolbeault complex with weights according to normal crossings, which is a useful tool for studying the -equation on singular complex spaces by resolution of singularities (where normal crossings appear naturally). The major difficulty is to prove that this complex is locally exact. We do that by constructing a local -solution operator which involves only Cauchy’s Integral Formula (in one complex variable) and behaves well for L ^p -forms with weights according to normal crossings.      

Approximate Identities, Almost-Periodic Functions and Toeplitz Operators

A sequence {A } of linear bounded operators is called stable if, for all sufficiently large , the inverses of A exist and their norms are uniformly bounded. We consider the stability problem for sequences of Toeplitz operators {T(k a)}, where a(t) is an almost-periodic function on unit circle and k a is an approximate identity. A stability criterion is established in terms of the invertibility of a family of almost-periodic functions. This family of functions depends on the approximate identity used in a very subtle way, and the stability condition is, in general, stronger than the invertibility condition of the Toeplitz operator T(a).     

Enhancement of the algebraic precision of a linear operator and consequences under positivity

Let Ω be a compact convex domain in and let L be a bounded linear operator that maps a subspace of C(Ω) into C(Ω). Suppose that L reproduces polynomials up to degree m. We show that for appropriately defined coefficients a_mrj the operator