Robert Sheckley’s Answerer for two orthogonal projections

Abstract

This paper is the written version of our talk (presented by the second author) at the IWOTA in Chemnitz in August 2017. The meta theorem of the paper is that Halmos' two projections theorem is something like Robert Sheckley's Answerer: no question about the W * -and C * -algebras generated by two orthogonal projections will go unanswered, provided the question is not foolish. An alternative approach to questions about two orthogonal projections makes use of the supersymmetry equality introduced by Avron, Seiler, and Simon. A noteworthy insight of the paper reveals that the supersymmetric approach is nothing but Halmos in different language and hence an equivalent Answerer.

The norm attainment problem for functions of projections

“…Let A be given by (4). Then λ ∈ C is an eigenvalue of A if and only if the spectral measure of the set {x ∈ σ(H):…”

“…Theorem 3. Let A be the operator given by (4). Then A ∈ N if and only if either (i) max (j,k)∈Λ |a jk | ≥ √ λ max or (ii) max (j,k)∈Λ |a jk | < √ λ max and the spectral measure of Σ(A) is non-zero.…”

“…This note is in the spirit of our paper on Robert Sheckley's answerer for two orthogonal projections [4]. The meta theorem of that paper is that the two projections theorem of Halmos is something like Robert Sheckley's answerer: no question about the W * -and C * -algebras generated by two orthogonal projections will go unanswered, provided the question is not foolish.…”

Simultaneous extension of two bounded operators between Hilbert spaces

“…Thus, we wish to describe our results in the language of this far-reaching approach. We also refer the reader to the survey paper [4] for a detailed reference on these matters, and to [5] for even more examples. If M and N are two closed subspaces in a Hilbert space H, then the subspaces M and M ⊥ can be decomposed with respect to N as:…”

The norm attainment problem for functions of projections

“…This note is in the spirit of paper [4]. The meta theorem of that paper is that the two projections theorem of Halmos is something like Robert Sheckley's Answerer: no question about the W * -and C * -algebras generated by two orthogonal projections will go unanswered, provided the question is not foolish.…”