arXiv subjects

P Shankar

Publications and source records attributed to P Shankar.

2 recordsLinked to original sources

arXiv2024

Isometric pairs with compact and normal cross-commutator

We represent and classify pairs of commuting isometries $(V_1, V_2)$ acting on Hilbert spaces that satisfy the condition \[ [V_1^*, V_2] = \text{compact and normal}, \] where $[V_1^*, V_2] := V_1^* V_2 - V_2 V_1^*$ is the cross-commutator of $(V_1, V_2)$. The precise description of such pairs also gives a complete and concrete set of unitary invariants. The basic building blocks of representations of such pairs consist of four distinguished pairs of commuting isometries. One of them relies on some peculiar examples of invariant subspaces tracing back to Rudin's intricate constructions of analytic functions on the bidisc. Along the way, we present a rank formula for a general pair of commuting isometries that looks to be the first of its kind.

math.FA↗

arXiv2022

On products of symmetries in von Neumann algebras

Let $\mathscr{R}$ be a type $II_1$ von Neumann algebra. We show that every unitary in $\mathscr{R}$ may be decomposed as the product of six symmetries (that is, self-adjoint unitaries) in $\mathscr{R}$, and every unitary in $\mathscr{R}$ with finite spectrum may be decomposed as the product of four symmetries in $\mathscr{R}$. Consequently, the set of products of four symmetries in $\mathscr{R}$ is norm-dense in the unitary group of $\mathscr{R}$. Furthermore, we show that the set of products of three symmetries in a von Neumann algebra $\mathscr{M}$ is not norm-dense in the unitary group of $\mathscr{M}$. This strengthens a result of Halmos-Kakutani which asserts that the set of products of three symmetries in $\mathcal{B}(\mathscr{H})$, the ring of bounded operators on a Hilbert space $\mathscr{H}$, is not the full unitary group of $\mathcal{B}(\mathscr{H})$.

math.OA↗