We show that, in complex interpolation, an operator function that is compact on one side of the interpolation scale will be compact for all proper interpolating spaces if the right hand side (Y , Y ) is reduced to a single space. A corresponding result, in restricted generality, is shown if the left hand side (X, X) is reduced to a single space. These results are derived from the fact that a ho...

Let F be an arbitrary topological vector space; we shall say that a subset S of F is quasi-convex if the set of continuous affine functionals on 5 separates the points of S. If X is a Banach space and T: X -* F is a continuous linear operator, then T is quasi-convex if T(U) is quasiconvex, where U is the unit ball of X. In the case when T is compact, T(U) is quasi-convex if and only if it is af...

Let D ⊂ R be a compact convex subset with nonempty interior. If A : C(D)→ C(D) is a positive linear operator with Π0(D) ⊂ FA or Π1(D) ⊂ FA then we establish some relations between the mixed-extremal point set of D and the interpolation point set of A. Our results include some well known results (see I. Raşa, Positive linear operators preserving linear functions, Ann. T. Popoviciu Seminar of Fun...

For a bounded linear operator on Hilbert space we define a sequence of the so-called weakly extremal vectors‎. ‎We study the properties of weakly extremal vectors and show that the orthogonality equation is valid for weakly extremal vectors‎. ‎Also we show that any quasinilpotent operator $T$ has an hypernoncyclic vector‎, ‎and so $T$ has a nontrivial hyperinvariant subspace‎.

; The Pade approximant problem is related to a (not necessarily orthogonal) projection of a linear functional equation of the Fredholni type. If the kernel is of trace class and its upper Hessenberg form is tridiagonal (this class includes Hermitian operators), then we prove that not only do the diagonal Pade' appioximants converge, but so do their numerators and denominators separately. The ge...

In this paper we show that harmonic compact operator-valued functions are characterized by having harmonic diagonal matrix coefficients in any choice of basis. We also give an example which shows that an operator-valued function with values outside the compact operators can have harmonic diagonal matrix coefficients in any choice of basis without being a harmonic operator-valued function. We us...