Calculus of Operators: Covariant Transform and Relative Convolutions

Kisil, Vladimir V.

doi:10.15352/bjma/1396640061

Cited by 32 publications

(20 citation statements)

References 47 publications

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“…Induced Wavelet (Covariant) Transform. The following object is common in quantum mechanics [59], signal processing, harmonic analysis [76], operator theory [73,75] and many other areas [69]. Therefore, it has various names [2]: coherent states, wavelets, matrix coefficients, etc.…”

Section: 1mentioning

confidence: 99%

“…with a kernel k defined on X = G/H. There are many important classes of operators described by (138), notably pseudodifferential operators (PDO) and Toeplitz operators [36,55,56,75]. Thus, it is important to have various norm estimations of ρ(k).…”

Section: 5mentioning

confidence: 99%

“…Convolutions. If G is the Heisenberg group and ρ is its Schrödinger representation, then ρ(â) (138) is a PDO a(X, D) with the symbol a [26,36,75], which is the Weyl quantization (6) of a classical observable a defined on phase space R 2 . Here,â is the Fourier transform of a, as usual.…”

Section: Norm Estimations Of Relativementioning

confidence: 99%

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Symmetry, Geometry and Quantization with Hypercomplex Numbers

Kisil¹

2017

GIQ

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These notes describe some links between the group SL 2 (R), the Heisenberg group and hypercomplex numbers -complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this framework. In particular, classical mechanics can be obtained as a theory with noncommutative observables and a non-zero Planck constant if we replace complex numbers in quantum mechanics by dual numbers. Our consideration is based on induced representations which are build from complex-/dual-/double-valued characters. Dynamic equations, rules of additions of probabilities, ladder operators and uncertainty relations are discussed. Finally, we prove a Calderón-Vaillancourt-type norm estimation for relative convolutions.

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Section: 1mentioning

confidence: 99%

Section: 5mentioning

confidence: 99%

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Symmetry, Geometry and Quantization with Hypercomplex Numbers

Kisil¹

2017

GIQ

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“…Over the last decades, abstract non-commutative harmonic analysis has achieved a significant popularity in coherent state transforms such as timescale (wavelet) transform and time-frequency (Gabor) transform and continuous frame theory, see [3,[8][9][10][11][21][22][23]26] and standard references therein.…”

Section: Introductionmentioning

confidence: 99%

Operator-Valued Continuous Gabor Transforms over Non-unimodular Locally Compact Groups

Farashahi

2017

Mediterr. J. Math.

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Abstract. In this article, we present the abstract harmonic analysis aspects of the operator-valued continuous Gabor transform (CGT) on second countable, non-unimodular, and type I locally compact groups. We show that the operator-valued continuous Gabor transform CGT satisfies a Plancherel formula and an inversion formula. As an example, we study these results on the continuous affine group.Mathematics Subject Classification. Primary 43A30, 43A32, 43A65, 22D10.

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“…Characterizations of nuclear operators in terms of decomposition of symbol through Fourier transform were investigated by Ghaemi, Jamalpour Birgani and Wong for S 1 [18]. Later they generalized their results on nuclearity to the pseudo-differential operators for any arbitrary compact group [19].The homogeneous spaces of abstract compact groups play an important role in mathematical physics, geometric analysis, constructive approximation and coherent state transform, see [23,24,25,26,27,28] and the references therein. The study of pseudo-differential operators on homogeneous spaces of compact groups was started by the first author [29].…”

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confidence: 99%

Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups

Kumar

2018

Forum Mathematicum

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Given a compact (Hausdorff) group G and a closed subgroup H of G, in this paper we present symbolic criteria for pseudo-differential operators on compact homogeneous space G/H characterizing the Schatten-von Neumann classes Sr(L 2 (G/H)) for all 0 < r ≤ ∞. We go on to provide a symbolic characterization for r-nuclear, 0 < r ≤ 1, pseudo-differential operators on L p (G/H)-space with applications to adjoint, product and trace formulae. The criteria here are given in terms of the concept of matrix-valued symbols defined on noncommutative analogue of phase space G/H × G/H. Finally, we present applications of aforementioned results in the context of heat kernels. be in Schatten-von Neumann class S r of operators on L 2 (G/H) for 0 < r ≤ ∞; (2) to find criteria for pseudo-differential operators from L p1 (G/H) into L p2 (G/H) to be r-nuclear, 0 < r ≤ 1, for 1 ≤ p 1 , p 2 < ∞; and (3) applications to find a trace formula and to provide criteria for heat kernel to be nuclear on L p (G/H). In order to do this, we will use the global quantization developed for compact homogeneous spaces as a non-commutative analogue of the Kohn-Nirenberg quantization of operators on R n .Recently, several researchers started a extensive research for finding the criteria for Schatten classes and r-nuclear operators in terms of symbols with lower regularity [2,12,15,40,41]. Ruzhansky and Delgado [12,14] successfully drop the regularity condition at least in their setting using the matrix-valued symbols instead of standard Kohn-Nirenberg quantization. Inspired by the work of Delgado and Ruzhansky, we present symbolic criteria for pseudo-differential operators on G/H to be Schatten class using the matrix-valued symbols defined on G/H × G/H, a noncommutative analogue of phase space. It is well known that in the setting of Hilbert spaces the class of r-nuclear operators agrees with the Schatten-von Neumann ideal of order r [35]. In general, for trace class operators on Hilbert spaces, the trace of an operator given by integration of its integral kernel over the diagonal is equal to the sum of its eigenvalues. However, this property fails in Banach spaces. The importance of r-nuclear operator lies in the work of Grothendieck, who proved that for 2/3-nuclear operators, the trace in Banach spaces agrees with the sum of all the eigenvalues with multiplicities counted. Therefore, the notion of r-nuclear operators becomes useful. Now, the question of finding good criteria for ensuring the r-nuclearity of operators arises but this has to be formulated in terms different from those on Hilbert spaces and has to take into account the impossibility of certain kernel formulations in view of Carleman's example [6] (also see [12]). In view of this, we will establish conditions imposed on symbols instead of kernels ensuring the r-nuclearity of the corresponding operators.The initiative of finding the necessary and sufficient conditions for a pseudo-differential operator defined on a group to be r-nuclear has been started by Delgado and Wong [16]. The main ingr...

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Calculus of Operators: Covariant Transform and Relative Convolutions

Cited by 32 publications

References 47 publications

Symmetry, Geometry and Quantization with Hypercomplex Numbers

Symmetry, Geometry and Quantization with Hypercomplex Numbers

Operator-Valued Continuous Gabor Transforms over Non-unimodular Locally Compact Groups

Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups

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