Spectral Shorted Operators

If $$\mathcal{H}$$ is a Hilbert space, $$\mathcal{S}$$ is a closed subspace of $$\mathcal{H},$$ and A is a positive bounded linear operator on $$\mathcal{H},$$ the spectral shorted operator $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence $$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes $$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to $$\mathcal{S}.$$ We characterize the left spectral resolution of $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that dim $${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional case and for non invertible operators.     

Remarks on Rawnsley’s $$\varvec{\varepsilon }$$ε -function on the Fock–Bargmann–Hartogs domains

In this paper, we mainly study a family of unbounded non-hyperbolic domains in $$\mathbb {C}^{n+m}$$, called Fock–Bargmann–Hartogs domains $$D_{n,m}(\mu )$$ ($$\mu >0$$) which are defined as a Hartogs type domains with the fiber over each $$z\in \mathbb {C}^{n}$$ being a ball of radius $$e^{-\frac{\mu }{2} {\Vert z\Vert }^{2}}$$. The purpose of this paper is twofold. Firstly, we obtain necessary and sufficient conditions for Rawnsley’s $$\varepsilon $$-function $$\varepsilon _{(\alpha ,g)}(\widetilde{w})$$ of $$\big (D_{n,m}(\mu ), g(\mu ;\nu )\big )$$ to be a polynomial in $$\Vert \widetilde{w}\Vert ^2$$, where $$g(\mu ;\nu )$$ is a Kähler metric associated with the Kähler potential $$\nu \mu {\Vert z\Vert }^{2} -\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$. Secondly, using above results, we study the Berezin quantization on $$D_{n,m}(\mu )$$ with the metric $$\beta g(\mu ;\nu )$$$$(\beta >0)$$.     

Maps preserving a certain product of operators

Let $$\mathcal {A}$$ be a standard operator algebra on a Banach space $$\mathcal {X}$$ with $$ \dim \mathcal {X}\ge 3$$. In this paper, we determine the form of the bijective maps $$\phi :\mathcal {A}\longrightarrow \mathcal {A}$$ satisfying $$\begin{aligned} \phi \left( \frac{1}{2}(AB^2+B^2A)\right) = \frac{1}{2}[\phi (A)\phi (B)^{2}+\phi (B)^{2}\phi (A)], \end{aligned}$$for every $$A,B \in \mathcal {A}$$.     

Sufficient conditions for local equivalence of mappings

Let $$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$ be $$C^{r+1}$$ mappings and let $$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$ , $$0\in Z$$ , $$m\le n$$ . We will show that if there exist a neighbourhood U of $$0\in {\mathbb {R}}^n$$ and constants $$C,C'>0$$ and $$k>1$$ such that for $$x\in U$$ $$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$ for any $$i\in \{1,\dots , m\}$$ and for any $$s \in \mathbf {\mathbb {N}}^n_0$$ such that $$|s|\le r$$ , then there exists a $$C^r$$ diffeomorphism $$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$ such that $$f=g\circ \varphi $$ in a neighbourhood of $$0\in {\mathbb {R}}^n$$ . By $$\nu (df)$$ we denote the Rabier function.     

Schatten class Hankel and $$\overline{\partial }$$ ∂ ¯ -Neumann operators on pseudoconvex domains in $$\mathbb {C}^{n}$$ C n

Monatshefte für Mathematik - Let $$\Omega $$ be a $$C^2$$ -smooth bounded pseudoconvex domain in $$\mathbb {C}^n$$ for $$n\ge 2$$ and let $$\varphi $$ be a holomorphic function on $$\Omega $$...     

A Tree-valued Markov Process Associated with an Admissible Family of Branching Mechanisms

Let E?R be an interval. By studying an admissible family of branching mechanisms{ψt,t ∈E} introduced in Li [Ann. Probab., 42, 41-79(2014)], we construct a decreasing Levy-CRT-valued process {Tt, t ∈ E} by pruning Lévy trees accordingly such that for each t ∈E, Tt is a ψt-Lévy tree. We also obtain an analogous process {Tt*,t ∈E} by pruning a critical Levy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of {Tt,t ∈E} at the ascension time A := inf{t ∈E;Tt is finite} can be represented by{Tt*,t∈E}.The results generalize those studied in Abraham and Delmas [Ann. Probab., 40, 1167-1211(2012)].     

Nonlinear Triple Product $$A^{*}B + B^{*}A$$ for Derivations on $$\ast$$ -Algebras

Mathematical Notes - Let $$\mathcal{A}$$ be a prime $$\ast$$ -algebra. In this paper, assuming that $$\Phi:\mathcal{A}\to\mathcal{A}$$ satisfies $$\Phi(A\diamond B \diamond C)=\Phi(A)\diamond B...     

On compact anisotropic Weingarten hypersurfaces in Euclidean space

Let $$\Omega \subset \mathbb {R}^n$$ be a bounded mean convex domain. If $$\alpha <0$$ , we prove the existence and uniqueness of classical solutions of the Dirichlet problem in $$\Omega $$ for the $$\alpha $$ -singular minimal surface equation with arbitrary continuous boundary data.     

Characterization of symmetrically $$\Delta $$ Δ -normed operator ideals which are interpolation spaces between Schatten–von Neumann ideals

Positivity - Let $$\mathcal {E}$$ be a symmetrically $$\Delta $$ -normed ideal in B(H). For $$1\le p&lt;\infty ,$$ $$q\ge 1,$$ we give a necessary and sufficient condition for $$\mathcal {E}$$...     

Characterization of distributive semigroup operations on the positive real numbers

Given any cancellative continuous semigroup operation $$\star $$ on the positive real numbers $$\mathbf {R}_+$$ with the ordinary topology, we completely characterize the set $$\mathcal {D}_\star (\mathbf {R}_+)$$ of all cancellative continuous semigroup operations on $$\mathbf {R}_+$$ which are distributed by $$\star $$ in terms of homeomorphism. As a consequence, we show that an arbitrary semigroup operation in $$\mathcal {D}_\star (\mathbf {R}_+)$$ is homeomorphically isomorphic to the ordinary addition $$+$$ on $$\mathbf {R}_+$$.     

Counter-examples to the Dunford–Schwartz pointwise ergodic theorem on $$\varvec{L^1+L^\infty }$$ L 1 + L ∞

Extending a result by Chilin and Litvinov, we show by construction that given any $$\sigma $$ -finite infinite measure space $$(\Omega ,\mathcal {A}, \mu )$$ and a function $$f\in L^1(\Omega )+L^\infty (\Omega )$$ with $$\mu (\{|f|>\varepsilon \})=\infty $$ for some $$\varepsilon >0$$ , there exists a Dunford–Schwartz operator T over $$(\Omega ,\mathcal {A}, \mu )$$ such that $$\frac{1}{N}\sum _{n=1}^N (T^nf)(x)$$ fails to converge for almost every $$x\in \Omega $$ . In addition, for each operator we construct, the set of functions for which pointwise convergence fails almost everywhere is residual in $$L^1(\Omega )+L^\infty (\Omega )$$ .     

The Liouville-type theorem for problems with nonstandard growth derived by Caccioppoli-type estimate

Let u be a nonnegative solution to the PDI $$-\,\mathrm{div} \mathcal {A}(x, u, \nabla u)\geqslant \mathcal {B}(x,u, \nabla u)$$ in $$\Omega $$, where $$\mathcal {A}$$ and $$\mathcal {B}$$ are differential operators with p(x)-type growth. As a consequence of the Caccioppoli-type inequality for the solution u, we obtain the Liouville-type theorem under some integral condition. We simplify the assumptions on functions $$ \mathcal {A}$$ and $$ \mathcal {B}$$, and we do not restrict the range of p(x) by the dimension n, therefore we can cover quite general family of problems.     

Partial Decidable Presentations in Hyperarithmetic

Siberian Mathematical Journal - We study the problem of the existence of decidable and positive $$\Pi_1^1$$ - and $$\Sigma_1^1$$ -numberings of the families of $$\Pi_1^1$$ - and $$\Sigma_1^1$$...     

Anti-derivable linear maps at zero on standard operator algebras

Acta Mathematica Hungarica - Let $$\mathcal{X}$$ be a complex Banach space with $$\dim \mathcal{X}\geq 2$$ , and $$\mathcal{A} \subseteq \mathcal{B}(\mathcal{X})$$ be a standard operator algebra....     

Problems of existence of order copies of $$\ell ^\infty $$ ℓ ∞ and $$L_p(\nu )$$ L p ( ν ) in some non-Banach Köthe spaces

Positivity - For a monotone Orlicz function $$\varPhi $$ taking only values 0 and $$\infty $$ , it is showed that in both cases, the s-homogeneous norm $$\Vert \cdot \Vert _{\varPhi ,s}$$ , if...     

A note on strong skew Jordan product preserving maps on von Neumann algebras

Periodica Mathematica Hungarica - Let $$\mathcal {A}$$ be a von Neumann algebra acting on the complex Hilbert space $$\mathcal {H}$$ and $$\Phi {:}\,\mathcal {A} \longrightarrow \mathcal {A}$$ be a...     

Diophantine approximation with prime variables and mixed powers

The Ramanujan Journal - Suppose that $$\lambda _1, \lambda _2, \lambda _3, \lambda _4$$ are positive real numbers and $$\frac{\lambda _1}{\lambda _2}$$ is irrational and algebraic. Let $$\mathcal...     

Menon-type identities with respect to sets of units

The Ramanujan Journal - Let $$n\ge 1$$ and let $$\mathbb {Z}_n^\star $$ be the group of units in the ring of residual classes modulo n, let $$m\ge 0$$ , $$k\ge 0$$ , $$m+k\ge 1$$ , $$u_1,\ldots...     

Description of the Structure of Irregularity Sets of Linear Differential Systems with a Linear Parameter

Differential Equations - We consider the class of linear parametric differential systems $$\dot {x}=\mu A(t)x $$ defined on the half-line $$t\geq 0 $$ , where $$\mu \in \mathbb {R} $$ is a...     

On the distribution of partial sums of irrational rotations

Periodica Mathematica Hungarica - For an irrational $$\alpha $$ , we investigate the sums $$\sum _{i=1}^n \left( \{i \alpha \} - \frac{1}{2} \right) $$ and $$\sum _{i=1}^n \left\{ \left( \{i \alpha...