On operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.     

On operators commuting with Toeplitz operators modulo the finite rank operators

Let X be a bounded linear operator on the Hardy space H² of the unit disk. We show that if is of finite rank for every inner function θ, then X=T_?+F for some Toeplitz operator T_? and some finite rank operator F on H². This solves a variant of an open question where the compactness replaces the finite rank conditions.     

On the self-inverse operators

Let T be an operator on a Hilbert space . The problem of computing of the norm of T, norm of selfcommutators of T, and the numerical radius of T are discussed in many papers and a number of textbooks. In this paper we determine the relationships between these values for self inverse operators and explain how we can determine any three of these (‖T‖, ‖[T*,T]‖, ‖{T*,T}‖, and the numerical radius of T) by knowing any one of them. Also, we find the spectrum of T*T,[T*,T] and {T*,T} in the case that T is self inverse and the spectrum of T*T is an interval. Finally, by giving some examples on automorphic composition operators, we show that these results make it possible to replace lengthy computation with quick ones. Research partially supported by the Shiraz university Research Council Grant No. 86-GR-SC-32.     

On the existence of wave operators for Dirac operators

It is shown that Kuroda's criterion for the existence of wave operators in the Schrödinger case is also valid for Dirac operators if the mass m0. If m=0 a similar but stronger condition is sufficient.Part of the author's doctoral thesis at the University of Munich, Germany.     

On operators which commute with analytic Toeplitz operators modulo the finite rank operators

It is shown that an operator on the Hardy space (or ) commutes with all analytic Toeplitz operators modulo the finite rank operators if and only if . Here is a finite rank operator, and in the case , is a sum of a rational function and a bounded analytic function, and in the case , is a bounded analytic function.

     

On power-bounded operators and operators satisfying a resolvent condition

We obtain, by means of a classification of the eigenvalues, local estimates for holomorphic. functions of a class of linear operators on a finite dimensional linear vector space. We apply these methods to find new proofs of some theorems ofKreiss andMorton, and in addition we give a local estimate of the powers of the inverse of any nonsingular operator in this class.     

On the bivariate Shepard-Lidstone operators

We propose a new combination of the bivariate Shepard operators (Coman and Trîmbi?a?, 2001 [2]) by the three point Lidstone polynomials introduced in Costabile and Dell’Accio (2005) [7]. The new combination inherits both degree of exactness and Lidstone interpolation conditions at each node, which characterize the interpolation polynomial. These new operators find application to the scattered data interpolation problem when supplementary second order derivative data are given (Kraaijpoel and van Leeuwen, 2010 [13]). Numerical comparison with other well known combinations is presented.     

On Kantorovich-Stieltjes operators

Let ν be a finite Borel measure on[0,1]The Kantorovich-Stieltjes polynomials are de-fined byK_n ν=(n+1)N_(k,n)(nN),where N_(k,n)(x)=x~k(1-x)~(n-k)(x[0,1],k=1,2,…,n)are the basic Bernsteinpolynomials and I_(k,n):=[k/(n+1),(k+1)/(n+1)](k=0,1,…,n;nN).We prove that the maximaloperator of the sequence(K_n)is of weak type and the sequence of polynomials(K_n ν)con-verges a.e.on[0,1]to the Radon-Nikodym derivative of the absolutely continuous part of     

On subnormal operators

LetS be a pure subnormal operator such thatC*(S), theC*-algebra generated byS, is generated by a unilateral shiftU of multiplicity 1. We obtain conditions under which 5 is unitarily equivalent toα + βU, α andβ being scalars orS hasC*-spectral inclusion property. It is also proved that if in addition,S hasC*-spectral inclusion property, then so does its dualT andC*(T) is generated by a unilateral shift of multiplicity 1. Finally, a characterization of quasinormal operators among pure subnormal operators is obtained.     

On Racah operators

As a generalization of the well-known Racah coefficients (defined for finite-dimensional representations of semisimple Lie groups), we introduce the notion of Racah operators for locally compact groups with “nice” dual space. In the case of the group PSL(2,?), these operators are explicitly indicated.     

On intertwining operators

LetB(H) denote the algebra of operators on the Hilbert spaceH into itself. GivenA,BB(H), defineC (A, B) andR (A, B):B(H)B(H) byC (A, B) X=AX–XB andR(A, B) X=AXB–X. Our purpose in this note is a twofold one. we show firstly that ifA andB ^*B (H) are dominant operators such that the pure part ofB has non-trivial kernel, thenC ⁿ (A, B) X=0, n some natural number, implies thatC (A, B)X=C(A ^*,B ^*)X=0. Secondly, it is shown that ifA andB ^* are contractions withC ₀ completely non-unitary parts, thenR ⁿ (A, B) X=0 for some natural numbern implies thatR (A, B) X=R (A ^*,B ^*)X=C (A, B ^*)X=C (A ^*,B) X=0. In the particular case in whichX is of the Hilbert—Schmidt class, it is shown that his result extends to all contractionsA andB.     

On log-hyponormal operators

LetTB(H) be a bounded linear operator on a complex Hilbert spaceH.TB(H) is called a log-hyponormal operator itT is invertible and log (TT ^*)log (T ^* T). Since log: (0, )(–,) is operator monotone, for 0<p1, every invertiblep-hyponormal operatorT, i.e., (TT ^*) ^p (T ^* T) ^p , is log-hyponormal. LetT be a log-hyponormal operator with a polar decompositionT=U|T|. In this paper, we show that the Aluthge transform is . Moreover, ifmeas ((T))=0, thenT is normal. Also, we make a log-hyponormal operator which is notp-hyponormal for any 0<p.This research was supported by Grant-in-Aid Research No. 10640185