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Factorizations and singular value estimates of operators with Gelfand-Shilov and Pilipović kernels. (English) Zbl 1458.47009

The paper presents several interesting results in different directions. Its main ingredients are Pilipović spaces, factorization algebras, the Schatten–von Neumann property, and pseudo-differential operators. Pilipović spaces are defined in terms of series of Hermite functions and powers of the harmonic oscillator, see [S. Pilipović, SIAM J. Math. Anal. 17, 477–484 (1986; Zbl 0604.46042)], and can be seen as a generalization of the Gelfand-Shilov spaces. An operator class \(\mathcal{M}\) is called a factorization algebra if every element of \(\mathcal{M}\) can be written as a product of two operators in the same class. The analysis is not trivial if \(\mathcal{M}\) does not contain the identity, cf.[R. Beals, Duke Math. J. 44, 45–57 (1977; Zbl 0353.35088)]. The Schatten–von Neumann property, defined in terms of singular value estimates, has been subject of intensive investigation in these last years. The pseudo-differential operators considered here are those in the Gelfand-Shilov setting, cf. [M. Cappiello and J. Toft, Math. Nachr. 290, No. 5–6, 738–755 (2017; Zbl 1377.47017)]. The layout of the paper is, in short, the following. In the frame of Pilipović spaces, kernel theorems for operators are proved. Factorization properties are then obtained in the corresponding algebra. In turn, factorization properties are used to obtain singular value estimates.

MSC:

47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
47A58 Linear operator approximation theory
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46F05 Topological linear spaces of test functions, distributions and ultradistributions
47G30 Pseudodifferential operators
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References:

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