Fields of primary interest: linear operators, Banach algebras, complex analysis, algebra, history of mathematics.

The main results of over 70 publications (partly joint with 30 co-authors from 17 countries) concern:

Spectral characterizations of two-sided ideals and the centre of a Banach algebra clarifying, in particular, which quasi-nilpotent elements belong to the radical (stability by quasi-nilpotent perturbations, or local Lipschitz continuity of the spectral radius), and characterizing commutativity modulo the radical by properties of the spectral radius (subadditivity, submultiplicativity, uniform continuity).

The analytic structure of the set of idempotents and relations to the centre and quasi-nilpotent elements of a Banach algebra. Invariant subspaces for pairs of projections. Lifting of idempotents.

Characterizations of traces on operator algebras. Numerical ranges and Gerschgorin discs.

An identity theorem for countable-valued analytic functions.

The geometric genesis of various spectral values as limits of the form
lim_{n→∞}g(T^{n})^{1/n}, where g(T) is a suitable geometric
characteristic of the Banach space operator T. Localization of the spectrum.

Optimal perturbations: given a Banach space operator T and a compact subset
K of its semi-Fredholm domain, one can construct a finite rank operator F
such that T+F-λ is bounded from below or surjective for each
λ in K, and (FT-TF)^{2}=0.

Ergodic theory of linear operators. Operator semigroups. Resolvent conditions. Volterra
operators (e.g., the primitive function): the operator I-Volterra on
L^{p}(0,1), 1≤p≤∞, is
power-bounded if and only if p=2.

** Other activities:** Editorial Board of Studia Mathematica,
Mathematica Slovaca, Mathematical Proceedings of the Royal Irish Academy,
Journal of Mathematics and Applications, Filomat, Czechoslovak Mathematical
Journal.
Reviewer for
Mathematical Reviews and Zentralblatt für Mathematik. Banach Centre
books and meetings, hundreds of international conferences. Operator Theory
Seminar at IM PAN (1994-).

Project Coordinator of the program
"Operator Theory Methods for
Differential Equations (TODEQ)", 2006-2010, within the Marie Curie
Actions (Transfer of Knowledge) of the European Commission.

Supervisor of 7 PhD theses.

** Awards: **International Mathematical Olympiads, 1963 and 1964; Polish
Academy of Sciences, 1984 and 1985; Banach Prize of the Polish Mathematical
Society, 1987.