arXiv subjects

Sankar T. R

Publications and source records attributed to Sankar T. R.

4 recordsLinked to original sources

arXiv2025

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

Pairs of projections and commuting isometries

It is known that the non-zero part of compact defect operators of Berger-Coburn-Lebow pairs (BCL pairs in short) of isometries are diagonal operators of the form \[ \begin{bmatrix} I_1 & & & \\ & D & & \\ & & - I_2 & \\ & & & - D \\ \end{bmatrix}, \] where $I_1$ and $I_2$ are the identity operators and $D$ is a positive contractive diagonal operator. We discuss the question of constructing an irreducible BCL pair from a diagonal operator of the above type. The answer to this question is sometimes in the affirmative and sometimes in the negative. This also answers a part of the question raised by He, Qin, and Yang. Our explicit constructions of BCL pairs yield concrete examples of pairs of commuting isometries.

math.FA↗

arXiv2018

Pairs of commuting isometries - I

We present an explicit version of Berger, Coburn and Lebow's classification result for pure pairs of commuting isometries in the sense of an explicit recipe for constructing pairs of commuting isometric multipliers with precise coefficients. We describe a complete set of (joint) unitary invariants and compare the Berger, Coburn and Lebow's representations with other natural analytic representations of pure pairs of commuting isometries. Finally, we study the defect operators of pairs of commuting isometries.

math.FA↗

arXiv2017

Characterization of Invariant subspaces in the polydisc

We give a complete characterization of invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on the Hardy space $H^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$, $n >1$. In particular, this yields a complete set of unitary invariants for invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on $H^2(\mathbb{D}^n)$, $n > 1$. As a consequence, we classify a large class of $n$-tuples, $n > 1$, of commuting isometries. All of our results hold for vector-valued Hardy spaces over $\mathbb{D}^n$, $n > 1$. Our invariant subspace theorem solves the well-known open problem on characterizations of invariant subspaces of the Hardy space over the unit polydisc.

math.FA↗