Reducing subspaces of multiplication operators with the symbol αz k + βw l on $$L_a^2 (\mathbb{D}^2 )$$

Wang, XuDi; Dan, Hui; Huang, Hansong

doi:10.1007/s11425-015-4973-9

Cited by 27 publications

(7 citation statements)

References 22 publications

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“…If β = γ = −1, then H = D β ⊗ D γ is the Bergman space L 2 a (D 2 ) over the bidisk. Some similar and tedious manipulations still yield that f a and g b are simple functions, hence we obtain many results again in [3,27]. This provides a good explanation for their different behavior on α.…”

Section: Proofsupporting

confidence: 69%

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Reducing subspaces of tensor products of weighted shifts

Guo

Wang

2015

Sci. China Math.

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A unilateral weighted shift A is said to be simple if its weight sequence {αn} satisfies ∇ 3 (α 2 n ) = 0 for all n 2. We prove that if A and B are two simple unilateral weighted shifts, then A ⊗ I + I ⊗ B is reducible if and only if A and B are unitarily equivalent. We also study the reducing subspaces of A k ⊗ I + I ⊗ B l and give some examples. As an application, we study the reducing subspaces of multiplication operators M z k +αw l on function spaces.

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Section: Proofsupporting

confidence: 69%

“…The idea of this proof comes from [27,Lemma 3.7]. If the statement was false for some r 1, then by Lemma 2.1,…”

Section: Proofmentioning

confidence: 99%

Reducing subspaces of tensor products of weighted shifts

Guo

Wang

2015

Sci. China Math.

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“…To optimize the performance of erbium-based integrated amplifier devices, various host materials and different fabrication methods have been studied to achieve high erbium-incorporation inside different compounds [7][8][9][10][11][12][13][14][15][16][17][18][19]. Additionally, unique waveguide geometries have also been proposed to maximize the interaction of the guided beams with the active layer [20][21][22][23][24][25]. Despite the recent progress in the field, there is still a relatively long way to go before erbium-based integrated devices can be established as fundamental active building blocks in the silicon photonics industry.…”

Section: Introductionmentioning

confidence: 99%

Erbium-doped hybrid waveguide amplifiers with net optical gain on a fully industrial 300 mm silicon nitride photonic platform

Rönn¹,

Zhang²,

Zhang³

et al. 2020

Opt. Express

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Recently, erbium-doped integrated waveguide devices have been extensively studied as a CMOS-compatible and stable solution for optical amplification and lasing on the silicon photonic platform. However, erbium-doped waveguide technology still remains relatively immature when it comes to the production of competitive building blocks for the silicon photonics industry. Therefore, further progress is critical in this field to answer the industry's demand for infrared active materials that are not only CMOS-compatible and efficient, but also inexpensive and scalable in terms of large volume production. In this work, we present a novel and simple fabrication method to form cost-effective erbium-doped waveguide amplifiers on silicon. With a single and straightforward active layer deposition, we convert passive silicon nitride strip waveguide channels on a fully industrial 300 mm photonic platform into active waveguide amplifiers. We show net optical gain over sub-cm long waveguide channels that also include grating couplers and mode transition tapers, ultimately demonstrating tremendous progress in developing cost-effective active building blocks on the silicon photonic platform.

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“…Intuitively, the graded structures of Hilbert spaces that Hilbert polynomials can be defined on should attract some attentions. Lately, motivated by [12], [25], [32], we find that graded structure can be very useful in operator theory.…”

Section: Introductionmentioning

confidence: 99%

The graded structure induced by operators on a Hilbert space

Guo¹,

Wang²

2018

J. Math. Soc. Japan

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In this paper we define a graded structure induced by operators on a Hilbert space. Then we introduce several concepts which are related to the graded structure and examine some of their basic properties. A theory concerning minimal property and unitary equivalence is then developed. It allows us to obtain a complete description of V * (M z k ) on any H 2 (ω). It also helps us to find that a multiplication operator induced by a quasi-homogeneous polynomial must have a minimal reducing subspace. After a brief review of multiplication operator M z+w on H 2 (ω, δ), we prove that the Toeplitz operator T z+w on H 2 (D 2 ), the Hardy space over the bidisk, is irreducible.

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Reducing subspaces of multiplication operators with the symbol αz k + βw l on $$L_a^2 (\mathbb{D}^2 )$$

Cited by 27 publications

References 22 publications

Reducing subspaces of tensor products of weighted shifts

Reducing subspaces of tensor products of weighted shifts

Erbium-doped hybrid waveguide amplifiers with net optical gain on a fully industrial 300 mm silicon nitride photonic platform

The graded structure induced by operators on a Hilbert space

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