A remark on complete positivity of elementary operators

We show that many of the features of the theory of hypercyclic and supercyclic operators extend to that of finitely hypercyclic/supercyclic operators. In particular, subnormal operators, Banach space isometries, and thereforeC ₁ contractions are not finitely supercyclic.     

A remark on numerical range of semi-hyponormal operators

Let U be the unilateral shift on ?². For any complex numbers α and β, put T = αU + βU* and S = T ². Then we show that the operator S is convexoid.     

The numerical range of elementary operators

For n-tuplesA=(A ₁,...,A _n ) andB=(B ₁,...,B _n ) of operators on a Hilbert spaceH, letR _A,B denote the operator onL(H) defined by . In this paper we prove that

Essential numerical range of elementary operators

Let and denote two -tuples of operators with and Let denote the elementary operators defined on the Hilbert-Schmidt class by We show that

Here is the essential numerical range, is the joint numerical range and is the joint essential numerical range.

     

A remark on the behaviour ofL p-multipliers and the range of operators acting onL p-spaces

In this paper, new results are obtained concerning the uniform approximation property (UAP) inL ^p-spaces (p≠2,1,∞). First, it is shown that the “uniform approximation function” does not allow a polynomial estimate. This fact is rather surprising since it disproves the analogy between UAP-features and the presence of “large” euclidian subspaces in the space and its dual. The examples are translation invariant spaces on the Cantor group and this extra structure permits one to replace the problem with statements about the nonexistence of certain multipliers in harmonic analysis. Secondly, it is proved that the UAP-function has an exponential upper estimate (this was known forp=1, ∞). The argument uses Schauder’s fix point theorem. Its precise behaviour is left unclarified here. It appears as a difficult question, even in the translation invariant context.     

A remark on multivariate projection operators

This paper is a recapitulation of the work of L. Szili and P. Vértesi [4] on multivariate Fourier series with triangular type partial sums. Namely, we give another proof for the corresponding lower estimation, which, in a way, is more direct than the previous one in [4].     

A remark on almost periodic transition operators

Results of Rosenblatt on almost periodic transition operators are extended to the reducible case.     

Orthogonality of the range and the kernel of some elementary operators

We prove the orthogonality of the range and the kernel of an important class of elementary operators with respect to the unitarily invariant norms associated with norm ideals of operators. This class consists of those mappings , , where is the algebra of all bounded Hilbert space operators, and , , , are normal operators, such that , and . Also we establish that this class is, in a certain sense, the widest class for which such an orthogonality result is valid. Some other related results are also given.

     

A remark on Schrödinger operators

We study the almost everythere convergence to the initial dataf(x)=u(x, 0) of the solutionu(x, t) of the two-dimensional linear Schrödinger equation Δu=i? _t u. The main result is thatu(x, t) →f(x) almost everywhere fort → 0 iff ∈H ^p (R²), wherep may be chosen <1/2. To get this result (improving on Vega’s work, see [6]), we devise a strategy to capture certain cancellations, which we believe has other applications in related problems.     

A remark on eigenvalues of certain positive operators

For a transition operator onC(X) with an invariant measure having global support, the point spectrum of unit modulus and the corresponding eigenfunctions are related to the action of a compact group on a quotient space ofX. Work supported in part by the National Science Foundation.     

A remark on the regularity of the ν-function of dirac operators

In this paper, we prove the regularity near the origin of the ν-function of Dirac operators associated with certain non-torsion free connections.     

A remark on quasianalytic vectors for a pair of anticommuting operators

We prove that two self-adjoint operators that anticommute on the dense invariant domain of their common quasianalytic vectors are strongly anticommuting. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii, Zhurnal, Vol. 49, No. 6. pp. 878–880, June, 1997.     

Norms of elementary operators

Let and , , be bounded linear operators acting on a separable Hilbert space . In this note, we prove that Moreover, we prove that there exists an operator with such that if and only if there exists a unitary such that