arXiv subjects

Santu Bera

Publications and source records attributed to Santu Bera.

3 recordsLinked to original sources

arXiv2025

A transference principle for involution-invariant functional Hilbert spaces

Let $σ: \mathbb C^d \rightarrow \mathbb C^d$ be an affine-linear involution such that $J_σ= -1$ and let $U, V$ be two domains in $\mathbb C^d.$ Let $ϕ: U \rightarrow V$ be a $σ$-invariant $2$-proper map such that $J_ϕ$ is affine-linear and let $\mathscr H(U)$ be a $σ$-invariant reproducing kernel Hilbert space of complex-valued holomorphic functions on $U.$ It is shown that the space $\mathscr H_ϕ(V):=\{f \in \mathrm{Hol}(V) : J_ϕ\cdot f \circ ϕ\in \mathscr H(U)\}$ endowed with the norm $\|f\|_ϕ:=\|J_ϕ\cdot f \circ ϕ\|_{\mathscr H(U)}$ is a reproducing kernel Hilbert space and the linear mapping $\varGamma_ϕ$ defined by $ \varGamma_ϕ(f) = J_ϕ\cdot f \circ ϕ,$ $f \in \mathrm{Hol}(V),$ is a unitary from $\mathscr H_ϕ(V)$ onto $\{f \in \mathscr H(U) : f = -f \circ σ\}.$ Moreover, a neat formula for the reproducing kernel $κ_ϕ$ of $\mathscr H_ϕ(V)$ in terms of the reproducing kernel of $\mathscr H(U)$ is given. The above scheme is applicable to symmetrized bidisc, tetrablock, $d$-dimensional fat Hartogs triangle and $d$-dimensional egg domain. Although some of these are known, this allows us to obtain an analog of von Neumann's inequality for contractive tuples naturally associated with these domains.

math.CV↗

arXiv2025

Dirichlet-type spaces of the unit bidisc and toral completely hyperexpansive operators

We discuss a notion, originally introduced by Aleman in one variable, of Dirichlet-type space $\mathcal D(μ_1,μ_2)$ on the unit bidisc $\mathbb D^2,$ with superharmonic weights related to finite positive Borel measures $μ_1,μ_2$ on $\overline{\mathbb D}.$ The multiplication operators $\mathscr M_{z_1}$ and $\mathscr M_{z_2}$ by the coordinate functions $z_1$ and $z_2,$ respectively, are bounded on $\mathcal D(μ_1,μ_2)$ and the set of polynomials is dense in $\mathcal D(μ_1,μ_2).$ We show that the commuting pair $\mathscr M_{z}=(\mathscr M_{z_1},\mathscr M_{z_2})$ is a cyclic analytic toral completely hyperexpansive $2$-tuple on $\mathcal D(μ_1,μ_2).$ Unlike the one variable case, not all cyclic analytic toral completely hyperexpansive pairs arise as multiplication $2$-tuple $\mathscr M_z$ on these spaces. In particular, we establish that a cyclic analytic toral completely hyperexpansive operator $2$-tuple $T=(T_1,T_2)$ satisfying $I-T^*_1 T_1-T^*_2T_2+T^*_1T^*_2T_1T_2=0$ and having a cyclic vector $f_0$ is unitarily equivalent to $\mathscr{M}_z$ on $\mathcal{D}(μ_1, μ_2)$ for some finite positive Borel measures $μ_1$ and $μ_2$ on $\overline{\mathbb{D}}$ if and only if $\ker T^*$, spanned by $f_0$, is a wandering subspace for $T$.

math.FA↗

arXiv2023

Dirichlet-type spaces of the bidisc and Toral $2$-isometries

We introduce and study Dirichlet-type spaces $\mathcal D(μ_1, μ_2)$ of the unit bidisc $\mathbb D^2,$ where $μ_1, μ_2$ are finite positive Borel measures on the unit circle. We show that the coordinate functions $z_1$ and $z_2$ are multipliers for $\mathcal D(μ_1, μ_2)$ and the complex polynomials are dense in $\mathcal D(μ_1, μ_2).$ Further, we obtain the division property and solve Gleason's problem for $\mathcal D(μ_1, μ_2)$ over a bidisc centered at the origin. In particular, we show that the commuting pair $\mathscr M_z$ of the multiplication operators $\mathscr M_{z_1},$ $\mathscr M_{z_2}$ on $\mathcal D(μ_1, μ_2)$ defines a cyclic toral $2$-isometry and $\mathscr M^*_z$ belongs to the Cowen-Douglas class ${\bf B}_1(\mathbb D^2_r)$ for some $r >0.$ Moreover, we formulate a notion of wandering subspace for commuting tuples and use it to obtain a bidisc analog of Richter's representation theorem for cyclic analytic $2$-isometries. In particular, we show that a cyclic analytic toral $2$-isometric pair $T$ with cyclic vector $f_0$ is unitarily equivalent to $\mathscr M_z$ on $\mathcal D(μ_1, μ_2)$ if and only if $\ker T^*,$ spanned by $f_0,$ is a wandering subspace for $T.$

math.FA↗