[Paper Review] Compressions and Pinchings

Jean-Christophe Bourin|ArXiv.org|Nov 22, 2002

Mathematics and Applications10 references21 citations

TL;DR

This paper establishes that any sequence of strict contractions can be realized as a pinching (block diagonal compression) of an operator A provided the essential numerical range of A contains the unit disk. Using properties of the essential numerical range and operator compression techniques, the authors prove that such pinching is possible via orthogonal decompositions and unitary equivalence, extending results on numerical ranges and diagonal sets in Hilbert space operators.

ABSTRACT

There exist operators $A$ such that : for any sequence of contractions $\{A_n\}$, there is a total sequence of mutually orthogonal projections $\{E_n\}$ such that $ΣE_nAE_n=\bigoplus A_n$.

Motivation & Objective

• To characterize when a sequence of strict contractions can be realized as a pinching (block diagonal compression) of a given operator.
• To investigate the role of the essential numerical range in determining the compressibility of operators.
• To extend known results on numerical ranges and diagonal sets to infinite-dimensional Hilbert spaces using compact perturbations and orthogonal decompositions.
• To resolve open questions about the convexity and connectivity of the diagonal set of operators.
• To establish conditions under which a sequence of normal operators can be realized as a pinching of a single operator.

Proposed method

• Uses the essential numerical range $W_e(A)$ defined as the intersection of closures of numerical ranges of $A+K$ over all compact operators $K$.
• Applies Parker’s theorem to redistribute diagonal entries via unitary equivalence, ensuring uniform distribution of values in the essential numerical range.
• Constructs an orthonormal basis such that the diagonal entries of $A$ along this basis are dense in a given compact convex set $\mathcal{D}$, using a recursive partitioning argument.
• Employs the Berg-Weyl-von Neumann theorem to handle non-diagonalizable normal operators by approximating them with diagonalizable ones.
• Uses a block decomposition of the Hilbert space into orthogonal subspaces $F_j$, each supporting a compression $A_{F_j}$ with essential numerical range containing $\mathcal{D}$.
• Applies a scaling and averaging argument to realize any strict contraction $X$ as a diagonal block via a finite sum of compressions, leveraging the uniform bound $\|A_n\|<1$.

Experimental results

Research questions

• RQ1 Under what conditions can a sequence of strict contractions be realized as a pinching of a single operator?
• RQ2 Can any normal operator with numerical range contained in the essential numerical range of another operator be realized as a compression?
• RQ3 Is the diagonal set $\Delta(A)$ of an operator always convex, and can it be disconnected for self-adjoint + compact operators?
• RQ4 What is the role of uniform norm bounds less than 1 in enabling pinching of sequences of contractions?
• RQ5 Can the essential numerical range be used to characterize the existence of orthogonal decompositions yielding prescribed block diagonal structures?

Key findings

• If $\sup_n\|A_n\|<1$ and $W_e(A)$ contains the unit disk, then there exists a pinching $\mathcal{P}(A) = \bigoplus A_n$ via a total sequence of orthogonal projections.
• For any strict contraction $X$, there exists an isometry $V$ such that $X = V^*AV$, provided $W_e(A) \supset \mathcal{D}$, the closed unit disk.
• For any contraction $X$, there exists a sequence $\{U_n\}$ of unitaries such that $U_n^*AU_n \to X$ in the weak operator topology.
• The strict inclusion $\bigcup_n W(A_n) \subset\subset W_e(A)$ is necessary and sufficient for the existence of a pinching $\mathcal{P}(A) = \bigoplus A_n$.
• The assumption $\|A_n\|<1$ uniformly cannot be dropped, as shown by a counterexample involving a halving projection and trace-class contradiction.
• The diagonal set $\Delta(A)$ satisfies $\mathrm{int}\,W_e(A) \subset \Delta(A) \subset W_e(A)$, and $\Delta(A)$ is symmetric about the real axis for real operators.