  For any measurable set $E$ of a measure space $(X, \mu)$, let $P_E$ be the
(orthogonal) projection on the Hilbert space $L^2(X, \mu)$ with the range $ran
\, P_E = \{f \in L^2(X, \mu) : f = 0 \ \ a.e. \ on \ E^c\}$ that is called a
standard subspace of $L^2(X, \mu)$. Let $T$ be an operator on $L^2(X, \mu)$
having increasing spectrum relative to standard compressions, that is, for any
measurable sets $E$ and $F$ with $E \subseteq F$, the spectrum of the operator
$P_E T|_{ran \, P_E}$ is contained in the spectrum of the operator $P_F T|_{ran
\, P_F}$. In 2009, Marcoux, Mastnak and Radjavi asked whether the operator $T$
has a non-trivial invariant standard subspace. They answered this question
affirmatively when either the measure space $(X, \mu)$ is discrete or the
operator $T$ has finite rank. We study this problem in the case of trace-class
kernel operators. We also slightly strengthen the above-mentioned result for
finite-rank operators.
