Multivariable isometries related to certain convex domains

Athavale, Ameer

doi:10.1216/rmj-2018-48-1-19

Cited by 2 publications

(3 citation statements)

References 43 publications

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“…The proof of Theorem 2.1 below is similar to the proofs of [4, Theorem 3.2] and [5, Proposition 4.6]; however, unlike there, it circumvents using the Taylor functional calculus of [19]. Also, unlike in [4] and [5], the Shilov boundary S Ω of Ω may not coincide with the topological boundary ∂Ω of Ω.…”

Section: A Lifting Theorem For Certain S ω -Isometriesmentioning

confidence: 87%

“…which is a lifting result for the intertwiner of toral isometries. In [5], the author introduced a class Ω (n) of convex domains Ω p in C n that satisfy the property (A); for n ≥ 2, the class Ω (n) happens to be distinct from the class of strictly pseudoconvex domains and the class of bounded symmetric domains in C n . Letting Ω to be Ω p , Theorem 2.1 (but not Corollary 2.2) captures [5,Proposition 4.6].…”

Section: A Lifting Theorem For Certain S ω -Isometriesmentioning

confidence: 99%

“…In [5], the author introduced a class Ω (n) of convex domains Ω p in C n that satisfy the property (A); for n ≥ 2, the class Ω (n) happens to be distinct from the class of strictly pseudoconvex domains and the class of bounded symmetric domains in C n . Letting Ω to be Ω p , Theorem 2.1 (but not Corollary 2.2) captures [5,Proposition 4.6]. A variant of Theorem 2.1 that is valid for (not necessarily convex) strictly pseudoconvex bounded domains Ω with C 2 boundary was proved in [4]; however, Theorem 2.1 does apply to strictly pseudoconvex bounded domains that are convex since any strictly pseudoconvex bounded domain Ω is known to satisfy the property (A) (refer to [1] and [9]).…”

Section: A Lifting Theorem For Certain S ω -Isometriesmentioning

confidence: 99%

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A note on Cartan isometries

Athavale¹

2019

Preprint

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We record a lifting theorem for the intertwiner of two S Ω -isometries which are those subnormal operator tuples whose minimal normal extensions have their Taylor spectra contained in the Shilov boundary of a certain function algebra associated with Ω, Ω being a bounded convex domain in C n containing the origin. The theorem captures several known lifting results in the literature and yields interesting new examples of liftings as a consequence of its being applicabile to Cartesian products Ω of classical Cartan domains in C n . Further, we derive intrinsic characterizations of S Ω -isometries where Ω is a classical Cartan domain of any of the types I, II, III and IV, and we also provide a neat description of an S Ω -isometry in case Ω is a finite Cartesian product of such Cartan domains. For a bounded domain Ω in C n , we let A(Ω) = {f ∈ C( Ω) : f is holomorphic on Ω}, where C( Ω) denotes the algebra of continuous functions on the closure Ω of Ω. The Shilov boundary of A(Ω) (or Ω) is defined to be the smallest closed subset S Ω of Ω such that, for any f ∈ A(Ω), sup{|f (z)| : z ∈ Ω} = sup{|f (z)| : z ∈ S Ω }.

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Section: A Lifting Theorem For Certain S ω -Isometriesmentioning

confidence: 87%

Section: A Lifting Theorem For Certain S ω -Isometriesmentioning

confidence: 99%

Section: A Lifting Theorem For Certain S ω -Isometriesmentioning

confidence: 99%

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A note on Cartan isometries

Athavale¹

2019

Preprint

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A-Isometries and Hilbert-A-Modules Over Product Domains

Didas

2022

Complex Anal. Oper. Theory

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For a compact set $$K \subset {\mathbb {C}}^n$$ K ⊂ C n , let $$A \subset C(K)$$ A ⊂ C ( K ) be a function algebra containing the polynomials $${\mathbb {C}}[z_1,\cdots ,z_n ]$$ C [ z 1 , ⋯ , z n ] . Assuming that a certain regularity condition holds for A, we prove a commutant-lifting theorem for A-isometries that contains the known results for isometric subnormal tuples in its different variants as special cases, e.g., Mlak (Studia Math. 43(3): 219–233, 1972) and Athavale (J. Oper. Theory 23(2): 339–350, 1990; Rocky Mt. J. Math. 48(1): 2018; Complex Anal. Oper. Theory 2(3): 417–428, 2008; New York J. Math. 25: 934–948, 2019). In the context of Hilbert-A-modules, our result implies the existence of an extension map "Equation missing" for hypo-Shilov-modules "Equation missing"$$(i=1,2)$$ ( i = 1 , 2 ) . By standard arguments, we obtain an identification "Equation missing" where "Equation missing" is the minimal $$C(\partial _A)$$ C ( ∂ A ) -extension of "Equation missing"$$(i=1,2)$$ ( i = 1 , 2 ) , provided that "Equation missing" is projective and "Equation missing" is pure. Using embedding techniques, we show that these results apply in particular to the domain algebra $$A=A(D)=C({\overline{D}})\cap {\mathcal {O}}(D)$$ A = A ( D ) = C ( D ¯ ) ∩ O ( D ) over a product domain $$D = D_1 \times \cdots \times D_k \subset {\mathbb {C}}^n$$ D = D 1 × ⋯ × D k ⊂ C n where each factor $$D_i$$ D i is either a smoothly bounded, strictly pseudoconvex domain or a bounded symmetric and circled domain in some $${\mathbb {C}}^{d_i}$$ C d i ($$1\le i \le k$$ 1 ≤ i ≤ k ). This extends known results from the ball and polydisc-case, Guo (Studia Math. 135(1): 1–12, 1999) and Chen and Guo (J. Oper. Theory 43: 69–81, 2000).

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Multivariable isometries related to certain convex domains

Cited by 2 publications

References 43 publications

A note on Cartan isometries

A note on Cartan isometries

A-Isometries and Hilbert-A-Modules Over Product Domains

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