The diagonal entries in the formula “quasitriangular $-$ compact $=$ triangular” and restrictions of quasitriangularity

Abstract:

A (bounded linear) Hilbert space operator $T$ is called quasitriangular if there exists an increasing sequence $\{ {P_n}\} _{n = 0}^\infty$ of finite-rank orthogonal projections, converging strongly to 1, such that $\left \| {(1 - {P_n})T{P_n}} \right \| \to 0 (n \to \infty )$. This definition, due to P. R. Halmos, plays a very important role in operator theory. The core of this article is a concrete answer to the following problem: Suppose $T$ is a quasitriangular operator and $\Gamma = \{ {\lambda _j}\} _{j = 1}^\infty$ is a sequence of complex numbers. Find necessary and sufficient conditions for the existence of a compact operator $K$ (of arbitrarily small norm) so that $T - K$ is triangular with respect to some orthonormal basis, and the sequence of diagonal entries of $T - K$ coincides with $\Gamma$. For instance, if no restrictions are put on the norm of $K$, then $T$ and $\Gamma$ must be related as follows: (a) if ${\lambda _0}$ is a limit point of $\Gamma$ and ${\lambda _0} - T$ is semi-Fredholm, then ${\operatorname {ind}}({\lambda _0} - T) > 0$; and (b) if $\Omega$ is an open set intersecting the Weyl spectrum of $T$, whose boundary does not intersect this set, then $\{ j:{\lambda _j} \in \Omega \}$ is a denumerable set of indices. Particularly important is the case when $\Gamma = \{ 0,0,0, \ldots \}$. The following are equivalent for an operator $T$: (1) there is an integral sequence $\{ {P_n}\} _{n = 0}^\infty$ of orthogonal projections, with rank ${P_n} = n$ for all $n$, converging strongly to 1, such that $\left \| {(1 - {P_n})T{P_{n + 1}}} \right \| \to 0 (n \to \infty )$; (2) from some compact $K, T - K$ is triangular, with diagonal entries equal to 0; (3) $T$ is quasitriangular, and the Weyl spectrum of $T$ is connected and contains the origin. The family ${({\text {StrQT}})_{ - 1}}$ of all operators satisfying (1) (and hence (2) and (3)) is a (norm) closed subset of the algebra of all operators; moreover, ${({\text {StrQT}})_{ - 1}}$ is invariant under similarity and compact perturbations and behaves in many senses as an analog of Halmos’s class of quasitriangular operators, or an analog of the class of extended quasitriangular operators ${({\text {StrQT}})_{ - 1}}$, introduced by the author in a previous article. If $\{ {P_n}\} _{n = 0}^\infty$ is as in (1), but condition $\left \| {(1 - {P_n})T{P_{n + 1}}} \right \| \to 0 (n \to \infty )$ is replaced by (1’) $\left \| {(1 - {P_{{n_k}}})T{P_{{n_k} + 1}}} \right \| \to 0 (k \to \infty )$ for some subsequence $\{ {n_k}\} _{k = 1}^\infty$, then (1’) is equivalent to (3’), $T$ is quasitriangular, and its Weyl spectrum contains the origin. The family ${({\text {QT}})_{ - 1}}$ of all operators satisfying (1’) (and hence (3’)) is also a closed subset, invariant under similarity and compact perturbations, and provides a different analog to Halmos’s class of quasitriangular operators. Both classes have “$m$-versions” (${({\text {StrQT}})_{ - m}}$ and, respectively, ${({\text {QT}})_{ - m}}$, $m = 1,2,3, \ldots$) with similar properties. (${({\text {StrQT}})_{ - m}}$ is the class naturally associated with triangular operators $A$ such that the main diagonal and the first $(m - 1)$ superdiagonals are identically zero, etc.) The article also includes some applications of the main result to certain nest algebras “generated by orthonormal bases.”