6th Summer Workshop on Operator Theory

9th - 13th July 2018 Kraków

Roman Drnovšek

Triangularizability of trace-class operators with increasing spectrum
For any measurable set E of a measure space (X, μ), let PE be the (orthogonal) projection on the Hilbert space L2(X, μ) with the range

ran PE = { f ∈ L2(X, μ) : f = 0 a.e. on  Ec}

that is called a standard subspace of L2(X, μ).
Let T be an operator on L2(X, μ) having increasing spectrum relative to standard compressions, that is, for any measurable sets E and F with E ⊆ F, the spectrum of the operator PE T|ran PF is contained in the spectrum of the operator PF T|ran PF.
The authors of [1] asked whether the operator T has a non-trivial invariant standard subspace.
They answered this question affirmatively when either the measure space (X, μ) is discrete or the operator T has finite rank. We study this problem in the case of trace-class kernel operators.

  1. L. W. Marcoux, M. Mastnak, H. Radjavi, Triangularizability of operators with increasing spectrum,
    J. Funct. Anal. 257 (2009), 3517-3540.
  2. R. Drnovšek , Triangularizability of trace-class operators with increasing spectrum, J. Math. Anal. Appl. 447 (2017), 1102-1115.
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