Jaroslav Zemánek

On the Gelfand-Hille theorems

Let T be a bounded linear operator on a complex Banach space X, with smallest possible spectrum, say, σ(T ) = {1}. Thus, the resolvent (T −λI)−1 is an analytic function of λ on C \ {1}, vanishing at infinity, and the point 1 is either a pole or an essential singularity. More precisely, it is a pole of order r if and only if r is the least exponent such that (T − I) = 0, because (T − λI)−1 = −I(λ− 1)−1− (T − I)(λ− 1)−2− . . .− (T − I)(λ− 1)−(n+1)− . . .

Characterizations of the trace

Abstract The paper contains a number of equivalent conditions which characterize the trace among the linear functionals on the matrix algebra. Some of these results are extended to more general operator algebras.

The stability radius of a semi-Fredholm operator

We prove an asymptotic formula for the stability radius of a semi-Fredholm operator on a Banach space in terms of the reduced minimum modulus. In particular, this gives a new proof of the Fredholm case studied by Förster and Kaashoek [4].

A characterization of normal matrices by their exponentials

Abstract We characterize normal matrices by logarithmic convexity of the singular values of their exponential groups.

Uniformly regular families of commuting operators

Let A(λ) be an analytic family of commuting n × n matrices with constant rank on an open connected set D. Then the range and the null space of Ak(λ) become independent of λ in D for a certain power k ⩽ n. We obtain similar results for uniformly regular families of commuting operators on a Banach space.

Gelfand-Hille theorems for cesàro means

The growth conditions on the powers in the Gelfand-Hille theorems can be replaced by the same conditions on the Cesàro means, retaining the conclusions. Moreover, the assumptions can be further weakened on one of the two sides.

The Gerschgorin discs under unitary similarity

The intersection of the Gerschgorin regions over the unitary similarity orbit of a given matrix is studied. It reduces to the spectrum in some cases: for instance, if the matrix satisfies a quadratic equation, and also for matrices having “large” singular values or diagonal entries. This leads to a number of open questions. 1. Motivation and the background. The unitary operators on finite-dimensional Hilbert spaces are distinguished among all linear operators by a number of remarkable properties. In particular, the unitary similarity orbit U(A) of a given operator A, that is, the set of operators of the form U∗AU , where U is unitary, can shed light on the operator A as well as on the structure of the group U of all unitary operators. There are signs indicating the richness of the unitary group U . For instance, the Schur triangularization theorem [I. Schur 1909] guarantees the existence of a triangular matrix form (with respect to a given orthonormal basis) of a suitable member of U(A); the diagonal of this matrix gives the spectrum of A. The idea can be traced back to some earlier sources cited already by Schur, and even to Cauchy and Jacobi; for the history see e.g. [E. T. Browne 1958], [T. Hawkins 1975], [C. C. MacDuffee 1946], [L. Mirsky 1955], [H. W. Turnbull and A. C. Aitken 1932]. The standard simple proof has now a permanent place in textbooks (cf. [R. A. Horn and C. R. Johnson 1985], [V. V. Prasolov 1994]). Also the singular values of A can be seen within the unitary similarity orbit U(A): there is a unitary U such that the rows of (the matrix representation with respect to an orthonormal basis of) U∗AU are orthogonal, and their Euclidean norms are the singular values of A (see [E. T. Browne 1928]). Every operator is a linear combination of four unitaries [A. G. Robertson 1974]. The whole algebra of operators is algebraically generated by two unitaries [C. Davis 1955], 1991 Mathematics Subject Classification: 15A18, 15A60. The paper is in final form and no version of it will be published elsewhere.

On polynomial connections between projections

Abstract We apply a result of Tremon to show that any two Banach-space projections of the same finite rank can be connected by a projection-valued polynomial path of degree not exceeding 3. Then we construct two similar infinite projections P and Q on a Hilbert space such that 1 is an eigenvalue of P′+Q′ for all projections P ′ and Q ′ with ‖ P − P ′‖ Q − Q ′‖

The reduced minimum modulus and the spectrum

We use the reduced minimum modulus to derive asymptotic formulae for the jumps of the minimum index of a semi-Fredholm operator on a Banach space. The results are then extended to the Riesz eigenvalues greater in absolute value than the essential spectral radius. This reveals a link between the two kinds of singularities.

Spectral classification of projections

Abstract Given two idempotents e and ƒ in a Banach algebra A , we study spectral characterizations to the effect that e and ƒ are not equivalent in A .