Title: THE INVARIANT SUBSPACE PROBLEM: GENERAL OPERATOR THEORY VS. CONCRETE OPERATOR THEORY?

Abstract: The Invariant Subspace Problem for (separable) Hilbert spaces is a long-standing open question that traces back to John Von Neumann’s works in the fifties asking, in particular, if every bounded linear operator acting on an infinite dimensional separable Hilbert space has a non-trivial closed invariant subspace. Whereas there are well-known classes of bounded linear operators on Hilbert spaces that have non-trivial, closed invariant subspaces (normal operators, compact operators, polynomially compact operators,...), the question of characterizing the lattice of the invariant subspaces of just a particular bounded linear operator is known to be extremely difficult and indeed, it may solve the Invariant Subspace Problem.

In this expository talk, we will focus on those concrete operators that may solve the Invariant Subspace Problem, presenting some of their main properties and exhibiting old and recent examples.

Professor Gallardo Gutierrez is giving a colloquium talk COMPACT PERTURBATIONS OF NORMAL OPERATORS: WHERE ARE THEIR INVARIANT SUBSPACES? at 3:30-4:20 p.m. the same of this talk and you are welcome to attend.