For each pair of numbers \(m,n\in {{\mathbb {N}}}\) with \(m>n\), we consider the norm on \({{\mathbb {R}}}^3\) given by \(\Vert (a,b,c)\Vert _{m,n}=\sup \{|ax^m+bx^{m-n}y^n+cy^m|:x,y\in [-1,1]\}\) for every \((a,b,c)\in {{\mathbb {R}}}^3\). We investigate some geometrical properties of these norms. We provide an explicit formula for \(\Vert \cdot \Vert _{m,n}\), a full description of the extreme points

We note that the well-known result of von Neumann (Contrib Theory Games 2:5–12, 1953) is not valid for all doubly substochastic operators on discrete Lebesgue spaces \(\ell ^p(I)\), \(p\in [1,\infty )\). This fact lead us to distinguish two classes of these operators. Precisely, the class of increasable doubly substochastic operators on \(\ell ^p(I)\) is isolated with the property that an analogue

In this paper, the numerical range of an even-order tensor is defined using the norm of its square matrix unfolding. The basic properties of the numerical range of a matrix, such as compactness and convexity, are proved to hold for the numerical range of an even-order tensor. Also, we introduce normal tensors based on the contraction product. According to the Tucker decomposition, we get the numerical

In this paper, we completely characterize the linear maps \(\phi :\mathcal {M} \rightarrow \mathcal {M}\) that preserve the Lorentz-cone spectrum, when \(\mathcal {M}\) is one of the following subspaces of the space \(M_{n}\) of \(n\times n\) real matrices: the subspace of diagonal matrices, the subspace of block-diagonal matrices \(\widetilde{A}\oplus [a]\), where \(\widetilde{A}\in M_{n-1}\) is symmetric

For vector lattices E and F, where F is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators \({\mathscr{L}}_{\mathrm{ob}}(E,F)\) from E into F. Using this, it follows that \({\mathscr{L}}_{\mathrm{ob}}(E,F)\) admits a Hausdorff uo-Lebesgue topology whenever F does. For each of order

We establish Leibniz type rules for bilinear flag multipliers with limited regularity in the Lebesgue spaces with flag weights. As applications, we obtain flag fractional Leibniz rules with weights, smoothing properties of bilinear flag fractional integrals, and scattering properties for solutions to a certain system of PDEs. Our results appear to be new even in the unweighted case.

Quite recently, a new property related to norm-attaining operators has been introduced: the weak maximizing property (WMP). In this note, we define a generalised version of it considering other topologies than the weak one (mainly the \(\hbox {weak}^*\) topology). We provide new sufficient conditions, based on the moduli of asymptotic uniform smoothness and convexity, which imply that a pair (X, Y)

The sample paths of white noise are proved to be elements of certain Besov spaces with dominating mixed smoothness. Unlike in isotropic spaces, here the regularity does not get worse with increasing space dimension. Consequently, white noise is actually much smoother than the known sharp regularity results in isotropic spaces suggest. An application of our techniques yields new results for the regularity

Let \(H_{p(\cdot ),q}\) be the variable Lorentz–Hardy martingale spaces. In this paper, we give a new atomic decomposition for these spaces via simple \(L_r\)-atoms \((1

For \(\mu , \beta \in {\mathbb {R}}\), we introduce and study in detail the generalized Stieltjes operators $$\begin{aligned} {\mathcal {S}}_{\beta ,\mu } f(t):={t^{\mu -\beta }}\int _0^\infty {s^{\beta -1}\over (s+t)^{\mu }}f(s)\mathrm{d}s, \qquad t>0, \end{aligned}$$ on Sobolev spaces \({{\mathcal{T}}_{p}^{{(\alpha )}}} (t^{\alpha })\) (where \(\alpha \ge 0\) is the fractional order of derivation

Let P(x, y) be a real-valued polynomial on \({\mathbb {R}}^n\times {\mathbb {R}}^n\). We denote by \(\deg _x(P)\) (resp., \(\deg _y(P)\)) the degree of P in x (resp., y). In this paper, we investigate the properties of the oscillatory integral given by \(T_{P,K}f(x)=\mathrm{p.v.}\int _{{\mathbb {R}}^n}\mathrm{e}^{iP(x,y)}K(x,y)f(y)\mathrm{d}y,\) where K is a Calderón–Zygmund non-convolutional type

In this paper we provide non-smooth atomic decompositions of 2-microlocal Besov-type and Triebel–Lizorkin-type spaces with variable exponents \(B^{\varvec{w}, \phi }_{p(\cdot ),q(\cdot )}({\mathbb {R}}^n)\) and \(F^{\varvec{w}, \phi }_{p(\cdot ),q(\cdot )}({\mathbb {R}}^n)\). Of big importance in general, and an essential tool here, are the characterizations of the spaces via maximal functions and

We study morphisms of the generalized quantum logic of tripotents in JBW\(^*\)-triples and von Neumann algebras. Especially, we establish a generalization of celebrated Dye’s theorem on orthoisomorphisms between von Neumann lattices to this new context. We show the existence of a one-to-one correspondence between the following maps: (1) quantum logic morphisms between the posets of tripotents preserving

We study embeddings of Morrey type spaces \({\varvec{M}}^{p,q,\omega } ({\mathbb {R}}^n ) \), \(1 \leqslant p<\infty \), \(1 \leqslant q<\infty \), both local and global, into weighted Lebesgue spaces \( {\varvec{L}}^p({\mathbb {R}}^n ,w) \), with the main goal to better understand the local behavior of functions \( f \in {\varvec{M}}^{p,q,\omega } ({\mathbb {R}}^n ) \) and also their behavior at infinity

A well-known application of the Ramsey Theorem is the proof of the Brunel–Sucheston Theorem. Based on this application, as an intermediate step, we consider the concept of \((k,\varepsilon )\)-oscillation stable sequence, which is generalized with the notion of \(((\mathcal {B}_i)_{i=1}^k,\varepsilon )\)-block oscillation stable sequence where \((\mathcal {B}_i)_{i=1}^k\) is a finite sequence of barriers

In this paper, we will use the convex modular \(\rho ^{*}(f)\) to investigate \(\Vert f\Vert _{\Psi ,q}^{*}\) on \((L_{\Phi })^{*}\) defined by the formula \(\Vert f\Vert _{\Psi ,q}^{*}=\inf _{k>0}\frac{1}{k}s_{q}(\rho ^{*}(kf))\), which is the norm formula in Orlicz dual spaces equipped with p-Amemiya norm. The attainable points of dual norm \(\Vert f\Vert _{\Psi ,q}^{*}\) are discussed, the interval

For a countably decomposable finite von Neumann algebra \({\mathscr {R}}\), we show that any choice of a faithful normal tracial state on \({\mathscr {R}}\) engenders the same measure topology on \({\mathscr {R}}\) in the sense of Nelson (J Funct Anal 15:103–116, 1974). Consequently it is justified to speak of ‘the’ measure topology of \({\mathscr {R}}\). Having made this observation, we extend the

In this paper, we study weighted composition–differentiation operators on the Hardy space \(H^{2}({\mathbb {D}})\). We investigate which combinations of weights \(\psi \) and maps of the open unit disk \(\varphi \) give rise to complex symmetric weighted composition–differentiation operators with conjugation \({\mathcal {C}}\), where \({\mathcal {C}}\) is a \(M_{z}\)-commuting conjugation on \(H^{2}({\mathbb

The present paper concerns the homogeneity and similarity of operators in Cowen-Douglas class \(B_n(\Omega )\). Let E be the Hermitian holomorphic vector bundle induced by \(T\in B_n(\mathbb {D})\), and \(E_{\alpha }\) be the Hermitian holomorphic vector bundle induced by \(\phi _{\alpha }(T)\), where \(\phi _{\alpha }\) is a M\(\ddot{o}\)bius transformation of the unit disk \(\mathbb {D}\). Assume

This work characterizes (dyadic homogeneous) wavelet frames for \(L^2({{\mathbb {R}}})\) by means of spectral techniques. These techniques use decomposability properties of the frame operator in spectral representations associated with the dilation operator. The approach is closely related to usual Fourier domain fiberization techniques, dual Gramian analysis, and extension principles. Spectral formulas

For an exotic locally compact Hausdorff space L, constructed under the assumption of Ostaszewski’s \(\clubsuit \)-principle, and a countable ordinal space \(\alpha \), we prove that all operators defined on \(C_0(\alpha \times L)\) have the simplest possible form. We also investigate the geometry of such space \(C_0(\alpha \times L)\) and we classify up to isomorphisms all its complemented subspaces

In this paper, we are interested in studying the set \(\mathcal {A}_{\Vert \cdot \Vert }(X, Y)\) of all norm-attaining operators T from X into Y satisfying the following: given \(\varepsilon >0\), there exists \(\eta \) such that if \(\Vert Tx\Vert > 1 - \eta \), then there is \(x_0\) such that \(\Vert x_0 - x\Vert < \varepsilon \) and T itself attains its norm at \(x_0\). We show that every norm one

Let \({\mathcal {M}}\) be a \(\sigma\)-finite von Neumann algebra, equipped with a normal faithful state \(\varphi\), and let \({\mathcal {A}}\) be maximal subdiagonal subalgebra of \({\mathcal {M}}\) and \(1\le p<\infty\). We prove a Beurling–Blecher–Labuschagne type theorem for \({{\mathcal {A}}}\)-invariant subspaces of Haagerup noncommutative \(L^p({{\mathcal {A}}})\) and give a characterization

We explore connections between boundary representations of operator spaces and those of the associated Paulsen systems. Using the notions of finite representation and separating property which we introduce for operator spaces, the boundary representations for operator spaces are characterized. We also introduce weak boundary for operator spaces. Rectangular hyperrigidity for operator spaces introduced

In this paper, we first establish the weighted compactness result for oscillation and variation associated with the truncated commutator of singular integral operators. Moreover, we establish a new \({{\mathrm{CMO}}({\mathbb {R}}^{n})}\) characterization via the compactness of oscillation and variation of commutators on weighted Lebesgue spaces.

We demonstrate how to construct spectral triples for twisted group \(C^*\)-algebras of lattices in phase space of a second-countable locally compact abelian group using a class of weights appearing in time–frequency analysis. This yields a way of constructing quantum \(C^k\)-structures on Heisenberg modules, and we show how to obtain such structures using Gabor analysis and certain weighted analogues

Let [b, T] be the commutator generated by b and T, where \(b\in \mathrm {BMO}({\mathbb {R}}^{n})\) and T is a Calderón–Zygmund singular integral operator. In this paper, the authors establish some strong type and weak type boundedness estimates including the \(L\log L\) type inequality for [b, T] on the Herz-type spaces with variable exponent. Meanwhile, the similar results for the commutators \([b

We study the generalization of m-isometries and m-contractions (for positive integers m) to what we call a-isometries and a-contractions for positive real numbers a. We show that an operator satisfying a certain inequality in hereditary form is similar to a-contraction. This improvement of [9, Theorem I] is based on some Banach algebras techniques. We show that our operator classes are closely connected

In this paper, we study continuous frames from projective representations of locally compact abelian groups of type \(\widehat{G}\times G\). In particular, using the Fourier transform on locally compact abelian groups, we obtain a characterization of maximal spanning vectors. As an application, for G, a compactly generated locally Euclidean locally compact abelian group or a local field with odd residue

Reconstruction of signals by their Fourier (transform) samples is investigated in many mathematical/engineering problems such as the inverse Radon transform and optical diffraction tomography. This paper concerns on the reconstruction of spline-spectra signals in \(V(\hbox {sinc}_{a})\) by finitely many Fourier samples, where \(\hbox {sinc}_{a}\) is the generalized sinc function. There are two main

For certain families of compact subsets of the plane, the isomorphism class of the algebra of absolutely continuous functions on a set is completely determined by the homeomorphism class of the set. This is analogous to the Gelfand–Kolmogorov theorem for C(K) spaces. In this paper, we define a family of compact sets comprising finite unions of convex curves and show that this family has the ‘Gelfand–Kolmogorov’

Let \(\mathcal {M}_{X(\mathbb {R})}\) be the Banach algebra of all Fourier multipliers on a Banach function space \(X(\mathbb {R})\) such that the Hardy–Littlewood maximal operator is bounded on \(X(\mathbb {R})\) and on its associate space \(X'(\mathbb {R})\). For two sets \(\varPsi ,\varOmega \subset \mathcal {M}_{X(\mathbb {R})}\), let \(\varPsi _\varOmega\) be the set of those \(c\in \varPsi\)

For \(i=1,2\), let \(E_i\) be a reflexive Banach lattice over \({\mathbb {R}}\) with a certain parameter \(\lambda ^+(E_i)>1\), let \(K_i\) be a locally compact (Hausdorff) topological space and let \({\mathcal {H}}_i\) be a closed subspace of \({\mathcal {C}}_0(K_i, E_i)\) such that each point of the Choquet boundary \({\text {Ch}}_{{\mathcal {H}}_i} K_i\) of \({\mathcal {H}}_i\) is a weak peak point

Current work defines Schur representation of a bilinear operator \(T: H \times H \rightarrow H\), where H is a separable Hilbert space. Introducing the concepts of self-adjoint bilinear operators, ordered eigenvalues and eigenvectors, we prove that if T is compact, self-adjoint, and its eigenvalues are ordered, then T has a Schur representation, thus obtaining a spectral theorem for T on real Hilbert

In this paper we introduce a function for multilinear operators that can be considered as an extension of the so-called outer measure associated to a linear operator ideal. We prove that it allows to characterize the operators that belong to a closed surjective ideal of multilinear operators as those having measure equal to zero. We also obtain some interpolation formulas for this new measure. As a

Let $$\varOmega $$ be a proper open subset of $${\mathbb {R}}^n$$ and $$p(\cdot ):\varOmega \rightarrow (0,\infty )$$ a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the author introduces the variable Hardy space $$H^{p(\cdot )}(\varOmega )$$ on $$\varOmega $$ by the radial maximal function and then characterize the space $$H^{p(\cdot )}(\varOmega

A standard technique in infinite dimensional holomorphy, which produced several useful results, uses the Borel transform to represent linear functionals on certain spaces of multilinear operators between Banach spaces as multilinear operators. In this paper we develop a technique to represent linear functionals, as linear operators, on spaces of multilinear operators that are beyond the scope of the

We establish a more precise description of those surjective or bijective continuous linear operators preserving orthogonality between C$^*$-algebras. The new description is applied to determine all uniformly continuous one-parameter semigroups of orthogonality preserving operators on an arbitrary C$^*$-algebra. We prove that given a family $\{T_t: t\in \mathbb{R}_0^{+}\}$ of orthogonality preserving

We consider the commutators $[b,T]$ and $[b,I_{\rho}]$ on Orlicz-Morrey spaces, where $T$ is a Calderon-Zygmund operator, $I_{\rho}$ is a generalized fractional integral operator and $b$ is a function in generalized Campanato spaces. We give a necessary and sufficient condition for the boundedness of the commutators on Orlicz-Morrey spaces. To do this we prove the boundedness of generalized fractional

In this paper, we extend the notion of orthogonality to the general elements of an absolute matrix order unit space and relate it to the orthogonality among positive elements. We introduce the notion of a partial isometry in an absolute matrix order unit space. As an application, we describe the comparison of order projections. We also discuss finiteness of order projections.

In spite of the important applications of real selfadjoint operators and monotone operators, very few papers have dealt in depth with the properties of such operators. In the present paper, we follow A. Rhodius to define the spectrum $$\sigma _{\mathbb {F}}(T)$$ and the numerical range $$W_{\mathbb {F}}(T)$$ of a selfadjoint operator T acting on a Hilbert space H over the real/complex field $$\mathbb

In this article, via combining Riesz norms with Morrey norms, the authors introduce and study the so-called Riesz–Morrey space, which differs from the John–Nirenberg–Campanato space in subtracting integral means. These spaces provide a bridge connecting both Lebesgue spaces and Morrey spaces which prove to be the endpoint spaces of Riesz–Morrey spaces. Moreover, the authors introduce a block-type space

We consider truncated Toeplitz operators acting on infinite dimensional model spaces. We then describe the kernels and ranks of commutators of truncated Toeplitz operators with symbols induced by certain inner functions. Our results generalize recent results of Chen et al. [Oper Matrices (to appear)] to infinite dimensional model spaces.

Dual truncated Toeplitz operators on the orthogonal complement of the model space $$K_u^2(=H^2 \ominus uH^2)$$ with u nonconstant inner function are defined to be the compression of multiplication operators to the orthogonal complement of $$K_u^2$$ in $$L^2$$ . In this paper, we give a complete characterization of the commutant of dual truncated Toeplitz operator $$D_z$$ , and we even obtain the commutant

This article is concerned with the random dynamics of a wide class of non-autonomous, non-local, fractional, stochastic p-Laplacian equations driven by multiplicative white noise on the entire space $${\mathbb {R}}^N$$ . We first establish the well-posedness of the equations when the time-dependent non-linear drift terms have polynomial growth of arbitrary orders $$p,q\ge 2$$ . We then prove that the

The present work either extends or improves several results on lineability of differentiable functions and derivatives enjoying certain special properties. Among many other results, we show that there exist large algebraic structures inside the following sets of special functions: (1) The class of differentiable functions with discontinuous derivative on a set of positive measure, (2) the family of

Let $p(\cdot):\ \mathbb{R}^n\to(0,\infty]$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. Let $H_A^{p(\cdot)}(\mathbb{R}^n)$ be the variable anisotropic Hardy space associated with $A$ defined via the non-tangential grand maximal function. In this article, via the known atomic characterization of $H_

We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending the previous work by Langenbruch. As a consequence, we give very general conditions for these spaces to be nuclear. In particular, we obtain the corresponding results

In this paper, we define almost convex space. Let $$T:X\rightarrow Y$$ be a linear bounded operator. This paper shows that: (1) If X is almost convex and 2-strictly convex, Y is a Banach space, D(T) is closed, N(T) is an approximatively compact Chebyshev subspace of D(T) and R(T) is a 2-Chebyshev hyperplane of Y, then there exists a homogeneous selection $${T^\sigma }$$ of $${T^\partial }$$ such that

A quantum injective frame is a frame that can be used to distinguish density operators (states) from their frame measurements, and the frame quantum detection problem asks to characterize all such frames. This problem was recently settled in Botelho-Andrade et al. (Springer Proc Math Stat 255:337–352, 2017) and Botelho-Andrade et al. (J Fourier Anal Appl 25:2268–2323, 2019) mainly for finite or infinite

The main theorem of this paper establishes a necessary and sufficient condition for embedding Schramm spaces $$\varPhi BV$$ into Chanturiya classes $$V[\nu ]$$ . This result is new even for the classical spaces in the theory of Fourier series, namely, for the Wiener and the Salem classes. Furthermore, it provides a characterization of the embedding of Waterman classes $$\varLambda BV$$ into $$V[\nu

We study pairs $(U,L_0)$, where $U$ is a unitary operator in $H$ and $L_0\subset H$ is a closed subspace, such that $$ P_{L_0}U|_{L_0}:L_0\to L_0 $$ has a singular value decomposition. Abstract characterizations of this condition are given, as well as relations to the geometry of projections and pairs of projections. Several concrete examples are examined.

Let $T$ be a bounded quaternionic normal operator on a right quaternionic Hilbert space $\mathcal{H}$. We show that $T$ can be factorized in a strongly irreducible sense, that is, for any $\delta >0$ there exist a compact operator $K$ with $\|K\|< \delta$, a partial isometry $W$ and a strongly irreducible operator $S$ on $\mathcal{H}$ such that \begin{equation*} T = (W+K) S. \end{equation*} We illustrate

In this paper, three new algorithms are proposed for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition in a 2-uniformly convex and uniformly smooth Banach space. The algorithms are constructed around the $$\phi $$ -proximal mapping associated with cost bifunction. The first algorithm is designed with the prior knowledge of the Lipschitz-type constant of bifunction. This means

In this paper, we study the weighted composition operator on the Fock space $\mf$ of slice regular functions. First, we characterize the boundedness and compactness of the weighted composition operator. Subsequently, we describe all the isometric composition operators. Finally, we introduce a kind of (right)-anti-complex-linear weighted composition operator on $\mf$ and obtain some concrete forms such

In this paper, we show different inequalities for fractional-order differential operators. In particular, the Sobolev, Hardy, Gagliardo–Nirenberg, and Caffarelli–Kohn–Nirenberg-type inequalities for the Caputo, Riemann–Liouville, and Hadamard derivatives are obtained. In addition, we show some applications of these inequalities.

For a Banach space, the notion of a regular linear relation is introduced and studied. Also, the regular resolvent set for a closed linear relation is introduced and investigated. Certain characterizations of regular resolvents are obtained in terms of the gap metric between corresponding null spaces, and in terms of generalized resolvents of the linear relation itself, respectively.

In this paper, a framework of vector-valued white noise functionals has been constructed as a Gel’fand triple $${\mathcal {W}}_{\alpha }\otimes {\mathcal {E}} \subset \varGamma (H)\otimes K \subset ( {\mathcal {W}}_{\alpha }\otimes {\mathcal {E}})^*$$ . Base on the Gel’fand triple, a new notion of Wick product of vector-valued white noise functionals induced by a bilinear mapping $${\mathfrak {B}}:{\mathcal

In this paper the Sobolev regularity properties are investigated of the commutators of fractional maximal functions, both in the global and local case. Some new bounds for the derivatives of the above commutators will be established. As several applications, the boundedness for these operators in Sobolev spaces as well as the bounds of these operators on the Sobolev spaces with zero boundary values

A definition of relative discrete spectrum of noncommutative W*-dynamical systems is given in terms of the basic construction of von Neumann algebras, motivated from three perspectives: First, as a complementary concept to relative weak mixing of W*-dynamical systems. Second, by comparison with the classical (i.e., commutative) case. And, third, by noncommutative examples.