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Differences of weighted composition operators from Hardy space to weighted-type spaces on the unit ball

Ze-Hua Zhou, Yu-Xia Liang (2012)

Czechoslovak Mathematical Journal

In this paper, we limit our analysis to the difference of the weighted composition operators acting from the Hardy space to weighted-type space in the unit ball of ℂ N , and give some necessary and sufficient conditions for their boundedness or compactness. The results generalize the corresponding results on the single weighted composition operators and on the differences of composition operators, for example, M. Lindström and E. Wolf: Essential norm of the difference of weighted composition operators....

Discrete Spectrum of the Periodic Schrödinger Operator with a Variable Metric Perturbed by a Nonnegative Potential

M. Sh. Birman, V. A. Sloushch (2010)

Mathematical Modelling of Natural Phenomena

We study discrete spectrum in spectral gaps of an elliptic periodic second order differential operator in L2(ℝd) perturbed by a decaying potential. It is assumed that a perturbation is nonnegative and has a power-like behavior at infinity. We find asymptotics in the large coupling constant limit for the number of eigenvalues of the perturbed operator that have crossed a given point inside the gap or the edge of the gap. The corresponding asymptotics...

Double weighted commutators theorem for pseudo-differential operators with smooth symbols

Yu-long Deng, Zhi-tian Chen, Shun-chao Long (2021)

Czechoslovak Mathematical Journal

Let - ( n + 1 ) < m ≤ - ( n + 1 ) ( 1 - ρ ) and let T a ∈ ℒ ρ , δ m be pseudo-differential operators with symbols a ( x , ξ ) ∈ ℝ n × ℝ n , where 0 < ρ ≤ 1 , 0 ≤ δ < 1 and δ ≤ ρ . Let μ , λ be weights in Muckenhoupt classes A p , ν = ( μ λ - 1 ) 1 / p for some 1 < p < ∞ . We establish a two-weight inequality for commutators generated by pseudo-differential operators T a with weighted BMO functions b ∈ BMO ν , namely, the commutator [ b , T a ] is bounded from L p ( μ ) into L p ( λ ) . Furthermore, the range of m can be extended to the whole m ≤ - ( n + 1 ) ( 1 - ρ ) .

Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations

Johannes Sjöstrand (2009)

Annales de la faculté des sciences de Toulouse Mathématiques

In this work we continue the study of the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random perturbations, by treating the case of multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects.

Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics

Michael Hitrik, Karel Pravda-Starov (2013)

Annales de l’institut Fourier

For a class of non-selfadjoint h –pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we give a precise description of...

Eigenvalues of Hille-Tamarkin operators and geometry of Banach function spaces

Thomas Kühn, Mieczysław Mastyło (2011)

Studia Mathematica

We investigate how the asymptotic eigenvalue behaviour of Hille-Tamarkin operators in Banach function spaces depends on the geometry of the spaces involved. It turns out that the relevant properties are cotype p and p-concavity. We prove some eigenvalue estimates for Hille-Tamarkin operators in general Banach function spaces which extend the classical results in Lebesgue spaces. We specialize our results to Lorentz, Orlicz and Zygmund spaces and give applications to Fourier analysis. We are also...

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