arXiv subjects

Deepak Pradhan

Publications and source records attributed to Deepak Pradhan.

3 recordsLinked to original sources

arXiv2022

Characterizations of complex symmetric Toeplitz operators

We present complete characterizations of Toeplitz operators that are complex symmetric. This follows as a by-product of characterizations of conjugations on Hilbert spaces. Notably, we prove that every conjugation admits a canonical factorization. As a consequence, we prove that a Toeplitz operator is complex symmetric if and only if the Toeplitz operator is $S$-Toeplitz for some unilateral shift $S$ and the transpose of the Toeplitz operator matrix is equal to the matrix of the Toeplitz operator corresponding to the basis of the unilateral shift $S$. Also, we characterize complex symmetric Toeplitz operators on the Hardy space over the open unit polydisc. Our results answer the well known open question about characterizations of complex symmetric Toeplitz operators.

math.FA↗

arXiv2022

Partially isometric Toeplitz operators on the polydisc

A Toeplitz operator $T_φ$, $φ\in L^\infty(\mathbb{T}^n)$, is a partial isometry if and only if there exist inner functions $φ_1, φ_2 \in H^\infty(\mathbb{D}^n)$ such that $φ_1$ and $φ_2$ depends on different variables and $φ= \barφ_1 φ_2$. In particular, for $n=1$, along with new proof, this recovers a classical theorem of Brown and Douglas. \noindent We also prove that a partially isometric Toeplitz operator is hyponormal if and only if the corresponding symbol is an inner function in $H^\infty(\mathbb{D}^n)$. Moreover, partially isometric Toeplitz operators are always power partial isometry (following Halmos and Wallen), and hence, up to unitary equivalence, a partially isometric Toeplitz operator with symbol in $L^\infty(\mathbb{T}^n)$, $n > 1$, is either a shift, or a co-shift, or a direct sum of truncated shifts. Along the way, we prove that $T_φ$ is a shift whenever $φ$ is inner in $H^\infty(\mathbb{D}^n)$.

math.FA↗

arXiv2019

Commutant lifting and Nevanlinna-Pick interpolation in several variables

This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.

math.FA↗