Inner functions and spherical isometries

Didas, Michael; Eschmeier, Jörg

doi:10.1090/s0002-9939-2011-11034-7

Cited by 9 publications

(18 citation statements)

References 14 publications

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“…A recent result of the authors (Proposition 3.1 in [10]) shows that the following definition for general A-isometries is consistent with Prunaru's definition for spherical isometries.…”

Section: Toeplitz Operatorssupporting

confidence: 64%

“…the remarks following Proposition 3.1 in [10] for the case of spherical isometries), the following approximation results are immediate consequences of Lemma 2.3, Theorem 2.4 and Proposition 2.5.…”

Section: Proposition Every A(d)-isometry On a Relatively Compact Strmentioning

confidence: 72%

“…If the point spectrum of T is empty, then exactly as in the proof of Proposition 3.3 from [10] it follows that the norm and essential norm of every T -Toeplitz operator coincide. T has empty point spectrum, then for every T -Toeplitz operator X ∈ B(H ), the equality X = inf{ X − K : K ∈ K(H )} holds.…”

Section: Proposition (Prunaru) For a Regularmentioning

confidence: 80%

“…More precisely, a word-by-word repetition of the proof of Lemma 1.1 in [10] yields the following result.…”

Section: Definitionmentioning

confidence: 91%

“…As the above examples show (see also Theorem 3.5 in [10]), many interesting aspects of the theory of Toeplitz operators on classical Hardy spaces can be rediscovered in the context of multi-variable subnormal isometries. The role of the surface measure in the classical theory will then be played by a scalar spectral measure of the minimal normal extension for the underlying subnormal tuple.…”

Section: Introductionmentioning

confidence: 90%

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On the essential commutant of analytic Toeplitz operators associated with spherical isometries

Didas

Eschmeier

Everard

2011

Journal of Functional Analysis

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Let T ∈ B(H ) n be an essentially normal spherical isometry with empty point spectrum on a separable complex Hilbert space H , and let A T ⊂ B(H ) be the unital dual operator algebra generated by T . In this note we show that every operator S ∈ B(H ) in the essential commutant of A T has the form S = X + K with a T -Toeplitz operator X and a compact operator K. Our proof actually covers a larger class of subnormal operator tuples, called A-isometries, which includes for example the tuple T = (M z 1 , . . . , M z n ) ∈ B(H 2 (σ )) n consisting of the multiplication operators with the coordinate functions on the Hardy space H 2 (σ ) associated with the normalized surface measure σ on the boundary ∂D of a strictly pseudoconvex domain D ⊂ C n . As an application we determine the essential commutant of the set of all analytic Toeplitz operators on H 2 (σ ) and thus extend results proved by Davidson (1977) [6] for the unit disc and Ding and Sun (1997) [11] for the unit ball.

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“…A recent result of the authors (Proposition 3.1 in [10]) shows that the following definition for general A-isometries is consistent with Prunaru's definition for spherical isometries.…”

Section: Toeplitz Operatorssupporting

confidence: 64%

Section: Proposition Every A(d)-isometry On a Relatively Compact Strmentioning

confidence: 72%

Section: Proposition (Prunaru) For a Regularmentioning

confidence: 80%

“…More precisely, a word-by-word repetition of the proof of Lemma 1.1 in [10] yields the following result.…”

Section: Definitionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 90%

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On the essential commutant of analytic Toeplitz operators associated with spherical isometries

Didas

Eschmeier

Everard

2011

Journal of Functional Analysis

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Dual Toeplitz Operators on the Sphere Via Spherical Isometries

Didas

Eschmeier

2015

Integr. Equ. Oper. Theory

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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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Inner functions and spherical isometries

Cited by 9 publications

References 14 publications

On the essential commutant of analytic Toeplitz operators associated with spherical isometries

On the essential commutant of analytic Toeplitz operators associated with spherical isometries

Dual Toeplitz Operators on the Sphere Via Spherical Isometries

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

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