We prove that, if $f:\mathbb{R}^n\to\mathbb{R}$ satisfies Fréchet's functional equation \[ \Delta_h^{m+1}f(x)=0 \ \ \text{ for all }x=(x_1,\cdots,x_n),h=(h_1,\cdots,h_n) \in\mathbb{R}^n, \] and $f(x_1,\cdots,x_n)$ is not an ordinary algebraic polynomial in the variables $x_1,\cdots,x_n$, then $f$ is unbounded on all non-empty open set $U\subseteq \mathbb{R}^n$. Furthermore, the set $\overline{G(f)}^{\mathbb{R}^{n+1}}$ contains an unbounded open set.

In this paper, we establish a Cameron--Storvick type theorem with respect to the Gaussian process. We then use this theorem to obtain various integration formulas involving the transform, the $\diamond$-product and the first variation.

We prove equality between the Topological Stable Rank and the Bass Stable Rank for finitely generated projective left modules over a unital $C^{\ast}$-algebra. In order to do so, the concept of Stable Rank of a Hilbert module is introduced.

For every, possibly unbounded, multiplication operator in $L^p$-space, $p\in\,]0,\infty[$, on finite separable measure space we show that multicyclicity, multi-$*$-cyclicity, and multiplicity coincide. This result includes and generalizes Bram's much cited theorem from 1955 on bounded $*$-cyclic normal operators. It also includes as a core result cyclicity of the multiplication operator $M_z$ by the complex variable $z$ in $L^p(\mu)$ for every $\sigma$-finite Borel measure $\mu$ on $\mathbb{C}$. The concise proof is based in part on the result that the function $e^{-\left|z\right|^2}$ is a $*$-cyclic vector for $M_z$ in $C_0(\mathbb{C})$ and further in $L^p(\mu)$. We characterize topologically those locally compact sets $X\subset \mathbb{C}$, for which $M_z$ in $C_0(X)$ is cyclic.

Wiener--Hopf plus Hankel and Wiener--Hopf minus Hankel operators in both frameworks of standard and variable exponent Lebesgue spaces are considered in this paper. The main aim is to describe certain dependencies between the Fredholm property of some Wiener--Hopf operators acting between variable exponent Lebesgue spaces and the invertibility of Wiener--Hopf plus and minus Hankel operators on all the standard Lebesgue spaces. Different types of Fourier symbols will be used but special focus will be considered on the Wiener subclass of almost periodic matrix functions. In the first part of the paper we will give a survey of investigations on related results. This will be useful at the end of the paper to derive the above mentioned dependencies between the operators under study.

A bounded linear operator $T$ on a Banach space $X$ is called subspace-hypercyclic for a nonzero subspace $M$ if $orb\left( {T,x} \right) \cap M$ is dense in $M$ for a vector $x \in X$, where $orb (T,x)=\{T^nx: n=0,1,2,\cdots\}$. Similarly, the bounded linear operator $T$ on a Banach space $X$ is called subspace-supercyclic for a nonzero subspace $M$ if there exists a vector whose projective orbit intersects the subspace $M$ in a relatively dense set. In this paper we provide a Subspace-Supercyclicity Criterion and offer two equivalent conditions of this criterion. At the same time, we also characterize other properties of subspace-supercyclic operators.

Given a finite set $Y$ in a reflexive Banach space $F$ and a family $\mathcal{C}$ of closed subspaces of $F$, we study the problem of finding a subspace $V_0$ in $\mathcal{C}$ that best approximates the data $Y$ in the sense that $\sum_{f \in Y} d(f,V_0)= \min_{V \in \mathcal{C} }\sum_{f \in Y} d(f,V)$, where $d$ is the distance function on $F$. In this paper, we give necessary conditions and sufficient conditions over $\mathcal{C}$ for which such a best approximation exists. In particular, when $F$ has finite dimension a characterization on $\mathcal{C}$ is given.

It is shown that some immediate consequences of the spectral theorem provide refinements and extensions of the several well-known inequalities involving eigenvalues and traces of Hermitian matrices. We obtain bounds for the spread and condition number of a Hermitian matrix.

In this paper we study truncated moment problems for $J$-self-adjoint, $J$-skew-self-adjoint and $J$-unitary operators. Conditions of the solvability are given. Some canonical solutions of the moment problems are constructed. As a by-product, some extension results for $J$-skew-symmetric and $J$-isometric operators are obtained.

Let $\varphi$ be an analytic self-map of the open unit disk. It is given several sufficient conditions on $\varphi$ for which there is $u\in H^2 \ H^{\infty}$ such that the weighted composition operator $M_u C_\varphi$ on $H^2$ is bounded.

In this paper, we obtain formulas for the analogue of conditional Wiener integrals for the functional $F$ of the form $$ F(x) = \exp\Big\{ \int_{0}^{T} V(x(t)) dt \Big\}, \quad x\in C[0,T] $$ where $V: \Bbb{R} \rightarrow \Bbb{R}$ is a potential function. We then apply this formula to obtain several integration formulas for the functionals involving various potential functions which is used in quantum mechanics and other physical theories.

In this paper, the notions of rank$-k$ numerical range and $k-$spectrum of rectangular complex matrices are introduced. Some algebraic and geometrical properties are investigated. Moreover, for $\epsilon \gt 0,$ the notion of Birkhoff-James approximate orthogonality sets for $\epsilon$-higher rank numerical ranges of rectangular matrices is also introduced and studied. The proposed definitions yield a natural generalization of standard higher rank numerical ranges.

We consider Hankel operators $H_{\bar{f}}$ with anti-meromorphic symbols $\bar{f}$ and describe several spectral properties of these operators. Namely, we study their boundedness, compactness and Schatten class membership.

In this paper, we investigate the notion $\phi$-injectivity for Banach $A$-modules, where $\phi$ is a character on $A.$ We obtain some hereditary properties of $\phi$-injectivity for certain classes of Banach modules related to closed ideals. These results allow us to study $\phi$-injectivity of certain Banach $A$-modules in commutative case, specially $\ell^{1}$-semilattice algebras. As an application, we give an example of a non-injective Banach module which is $\phi$-injective for each character $\phi.$

Assume $T\in L(X)$ is a bounded linear operator on a Banach space $X$, and that $T_n$ is a restriction of $T$ on $R(T^n)=T^n(X)$. In general, almost nothing can be said concerning the relationship between the spectral properties of $T$ and $T_n$. However, under some conditions, it is shown that several spectral properties introduced recently are the same for $T$ and $T_n$.

We give a new definition for the full crossed product, respectively reduced crossed product, of a pro-$C^{\ast}$-algebra $A\left[ \tau _{\Gamma }\right] $ by an action $\alpha $ and, using these new definitions, we investigate some of their properties.

In this paper, some functional versions for operator entropies as well as for their related operator inequalities are investigated. The theoretical results obtained by our functional approach immediately imply those of operator version in a simple and nice way.

A bounded linear operator $T$ acting on a Banach space $X$ satisfies property $(aBw)$, a strong version of a-Weyl's theorem, if the complement in the approximate point spectrum $\sigma_{a}(T)$ of the upper semi-B-Weyl spectrum $\sigma_{USBW}(T)$ is the set of all isolated points of approximate point spectrum which are eigenvalues of finite multiplicity. In this paper we investigate the property $(aBw)$ in connection with Weyl type theorems. In particular, we show that $T$ satisfies property $(aBw)$ if and only if $T$ satisfies a-Weyl's theorem and $\sigma_{USBW}(T)=\sigma_{USW}(T)$, where $\sigma_{USW}(T)$ is the upper semi-Weyl spectrum of $T$. The preservation of property $(aBw)$ is also studied under commuting nilpotent, quasi-nilpotent, power finite rank or Riesz perturbations. The theoretical results are illustrated by some concrete examples.

In this paper, the authors establish some real-variable characterizations of Herz-type Hardy spaces with variable exponent.

$\newcommand{\UIN}{|\kern-1pt|kern-1pt|}$ In this paper, we show a further improvement of the integral Heinz mean inequality and prove \begin{eqnarray*} \frac{1}{2} \UIN{A^{2}X+2AXB+XB^{2}}\leq \frac{2}{t+2}\UIN{A^{2}X+tAXB+XB^{2}} \ \ \mbox{for all}\ \ t\in (-2, 2].\end{eqnarray*} Then we show some refinements of unitarily invariant norm inequalities, in particular we proved that: If $A, B, X \in M_{n}$ with $A$ and $B$ positive definite, and $f$, $g$ are two continuous functions on $(0,\infty)$ such that $h(x)= \frac{f(x)}{g(x)}$ is Kwong, then \begin{eqnarray*}\UIN{A^{\frac{1}{2}}(f(A)Xg(B)+g(A)Xf(B))B^{\frac{1}{2}}}\leq \frac{k}{2}\UIN{ A^{2}X+2AXB+XB^{2}}\end{eqnarray*} holds for any unitarily invariant norm, where $k=\max{\{\frac{f(\lambda)g(\lambda)}{\lambda}| \lambda \in \sigma(A)\cup \sigma(B)\}}$.