Decomposition of the Tensor Product of Two Hilbert Modules

Ghara, Soumitra; Misra, Gadadhar

doi:10.1007/978-3-030-43380-2_11

Cited by 4 publications

(4 citation statements)

References 15 publications

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“…\end{equation*}$$Clearly

B^{(t)}

is a sesqui‐analytic hermitian function for any real number

t

. It follows from [11, Lemma 6.1] that

B^{(t)}

is quasi‐invariant with the multiplier

c(u)=\overline{u}

. A direct computation shows that

$$\begin{align} B^{(t)}(\bm{z}, \bm{w})= \frac{d+1}{(1-{\left\langle \bm{z}, \bm{w}\right\rangle})^{t (d+1)+2}} \def\eqcellsep{&}\begin{pmatrix} 1-\sum _{j\ne 1}z_j\bar{w}_j&z_2\bar{w}_1&\cdots & z_d\bar{w}_1\\[3pt] z_1...

…”

Section: A Class Of Quasi‐invariant Kernelsmentioning

confidence: 99%

“…Proof The multiplication

d

‐tuple

\bm{M}

on the Hilbert space

\mathcal {H}_K(\mathbb {B}_d, \mathbb {C}^d)

is bounded if and only if there exists

c&gt;0

such that

\left(c^2-\langle {\bm{z}},\,{\bm{w}} \rangle \right)K(\bm{z}, \bm{w})

is nonnegative definite [11, Lemma 2.7(ii)].

\begin{matrix} (c^{2} - ⟨ z, w ⟩) {K false(bold-italicz, bold-italicw false) |}_{res {double-struckC}^{d} \otimes {scriptP}_{ℓ}} = & {c^{2} (a_{ℓ, 1} - a_{ℓ, 2}) - (a_{ℓ - 1, 1} - a_{ℓ - 1, 2})} {⟨ z, w ⟩}^{l - 1} \bar{bold-italicw} {bold-italicz}^{†} \\ + (c^{2} a_{ℓ, 2} - a_{ℓ - 1, 2}) ⟨ z, w \end{matrix}

…”

Section: Scriptufalse(dfalse)$\mathcal {U}(d)$ Homogeneous Operatorsmentioning

confidence: 99%

“…a_{\underline{s}} E_{\underline{s}}(\bm{z}, \bm{w})$. It is known that

\mathcal {K}_{\bm{a}}

is not only a positive definite kernel but also quasi‐invariant under

\mathbb {K}

, see [11, Proposition 2.3 and Proposition 6.2]. Indeed,

\mathcal {K}_{\bm{a}}

transforms according to the rule:

\begin{equation*} {k^{-1}}^{\dagger } \mathcal {K}_{\bm{a}}(k^{-1} \cdot \bm{z}, k^{-1} \cdot \bm{w}) \overline{k^{-1}} = \mathcal {K}_{\bm{a}} (\bm{z}, \bm{w}), \,\, k\in \mathbb {K}, \end{equation*}

where

\dagger

denotes the transpose of a matrix.…”

Section: Introductionmentioning

confidence: 99%

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Commuting tuple of multiplication operators homogeneous under the unitary group

Ghara,

Kumar,

Misra

et al. 2024

Journal of London Math Soc

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Let be the group of unitary matrices. We find conditions to ensure that a ‐homogeneous ‐tuple is unitarily equivalent to multiplication by the coordinate functions on some reproducing kernel Hilbert space , . We describe this class of ‐homogeneous operators, equivalently, nonnegative kernels quasi‐invariant under the action of . We classify quasi‐invariant kernels transforming under with two specific choice of multipliers. A crucial ingredient of the proof is that the group has exactly two inequivalent irreducible unitary representations of dimension and none in dimensions , . We obtain explicit criterion for boundedness, reducibility, and mutual unitary equivalence among these operators.

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“…\end{equation*}$$Clearly

B^{(t)}

is a sesqui‐analytic hermitian function for any real number

t

. It follows from [11, Lemma 6.1] that

B^{(t)}

is quasi‐invariant with the multiplier

c(u)=\overline{u}

. A direct computation shows that

$$\begin{align} B^{(t)}(\bm{z}, \bm{w})= \frac{d+1}{(1-{\left\langle \bm{z}, \bm{w}\right\rangle})^{t (d+1)+2}} \def\eqcellsep{&}\begin{pmatrix} 1-\sum _{j\ne 1}z_j\bar{w}_j&z_2\bar{w}_1&\cdots & z_d\bar{w}_1\\[3pt] z_1...

…”

Section: A Class Of Quasi‐invariant Kernelsmentioning

confidence: 99%

“…Proof The multiplication

d

‐tuple

\bm{M}

on the Hilbert space

\mathcal {H}_K(\mathbb {B}_d, \mathbb {C}^d)

is bounded if and only if there exists

c&gt;0

such that

\left(c^2-\langle {\bm{z}},\,{\bm{w}} \rangle \right)K(\bm{z}, \bm{w})

is nonnegative definite [11, Lemma 2.7(ii)].

\begin{matrix} (c^{2} - ⟨ z, w ⟩) {K false(bold-italicz, bold-italicw false) |}_{res {double-struckC}^{d} \otimes {scriptP}_{ℓ}} = & {c^{2} (a_{ℓ, 1} - a_{ℓ, 2}) - (a_{ℓ - 1, 1} - a_{ℓ - 1, 2})} {⟨ z, w ⟩}^{l - 1} \bar{bold-italicw} {bold-italicz}^{†} \\ + (c^{2} a_{ℓ, 2} - a_{ℓ - 1, 2}) ⟨ z, w \end{matrix}

…”

Section: Scriptufalse(dfalse)$\mathcal {U}(d)$ Homogeneous Operatorsmentioning

confidence: 99%

“…a_{\underline{s}} E_{\underline{s}}(\bm{z}, \bm{w})$. It is known that

\mathcal {K}_{\bm{a}}

is not only a positive definite kernel but also quasi‐invariant under

\mathbb {K}

, see [11, Proposition 2.3 and Proposition 6.2]. Indeed,

\mathcal {K}_{\bm{a}}

transforms according to the rule:

\begin{equation*} {k^{-1}}^{\dagger } \mathcal {K}_{\bm{a}}(k^{-1} \cdot \bm{z}, k^{-1} \cdot \bm{w}) \overline{k^{-1}} = \mathcal {K}_{\bm{a}} (\bm{z}, \bm{w}), \,\, k\in \mathbb {K}, \end{equation*}

where

\dagger

denotes the transpose of a matrix.…”

Section: Introductionmentioning

confidence: 99%

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Commuting tuple of multiplication operators homogeneous under the unitary group

Ghara,

Kumar,

Misra

et al. 2024

Journal of London Math Soc

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“…We begin with a lemma, which is a variant of [17,Lemma 4.14]. We include its proof for the sake of completeness.…”

Section: Proof Of Theorem 23 and Its Consequencesmentioning

confidence: 99%

Dirichlet-type spaces of the unit bidisc and toral 2-isometries

Bera,

Chavan,

Ghara

2024

Can. J. Math.-J. Can. Math.

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We introduce and study Dirichlet-type spaces $\mathcal D(\mu _1, \mu _2)$ of the unit bidisc $\mathbb D^2,$ where $\mu _1, \mu _2$ are finite positive Borel measures on the unit circle. We show that the coordinate functions $z_1$ and $z_2$ are multipliers for $\mathcal D(\mu _1, \mu _2)$ and the complex polynomials are dense in $\mathcal D(\mu _1, \mu _2).$ Further, we obtain the division property and solve Gleason’s problem for $\mathcal D(\mu _1, \mu _2)$ 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 $\mathcal D(\mu _1, \mu _2)$ defines a cyclic toral $2$ -isometry and $\mathscr M^*_z$ belongs to the Cowen–Douglas class $\mathbf {B}_1(\mathbb D^2_r)$ 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 $\mathcal D(\mu _1, \mu _2)$ for some $\mu _1,\mu _2$ if and only if $\ker T^*,$ spanned by $f_0,$ is a wandering subspace for $T.$

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The Relationship of the Gaussian Curvature with the Curvature of a Cowen-Douglas Operator

Ghara,

Misra

2023

Fields Institute Communications

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Decomposition of the Tensor Product of Two Hilbert Modules

Cited by 4 publications

References 15 publications

Commuting tuple of multiplication operators homogeneous under the unitary group

Commuting tuple of multiplication operators homogeneous under the unitary group

Dirichlet-type spaces of the unit bidisc and toral 2-isometries

The Relationship of the Gaussian Curvature with the Curvature of a Cowen-Douglas Operator

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