Deepak K. D | Searcharxiv

arXiv2024

Analytic primes, $M$-ideals, and $p$-sets in $H^\infty(\mathbb{D})$

We investigate the structure of $p$-sets, $M$-ideals, and a newly introduced notion of analytic primes in $H^\infty(\mathbb{D})$, where $H^\infty(\mathbb{D})$ denotes the Banach algebra of all bounded analytic functions on the open unit disc $\mathbb{D}$ in $\mathbb{C}$. We prove that $M$-ideals in $H^\infty(\mathbb{D})$ are analytic primes and are dense in the Hardy space. Outer functions play a key role in representing closed principal ideals in $H^\infty(\mathbb{D})$ that are $M$-ideals. Some of our results apply to the polydisc. The results presented in this paper offer some new perspectives on $H^\infty(\mathbb{D})$.

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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 $H^\infty(\mathbb{D}^n)$.

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