Wavelet and Linear Algebra

Abstract Let $N\in B(H)$ be a normal operator acting on a real or complex Hilbert space $H$. Define $N^\dagger:=N_1^{-1}\oplus 0:\mathcal{R}(N)\oplus \mathcal{K}(N)\rightarrow H$, where $N_1=N|_{\mathcal{R}(N)}$. Let the {\it fractional semigroup} $\mathfrak{F}r(W)$ denote the collection of all words of the form $f_1^\diamond f_2^\diamond \cdots f_k^\diamond~$ in which $~f_j \in L^\infty (W)~$ and $~f^\diamond~$ is either $~f~$ or $~f^\dagger$, where $f^\dagger=\chi_{ \{ f\neq 0 \}}/(f+\chi_{\{f=0\}})$ and $L^\infty(W)$ is a certain normed functional algebra of functions defined on $\sigma_\mathbb{F}(W)$, besides that, $W=W^* \in B(H)$ and $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$ indicates the underlying scalar field. The {\it fractional calculus} $(f_1^\diamond f_2^\diamond \cdots f_k^\diamond)(W)$ on $\mathfrak{F}r(W)$ is defined as $f_1^\diamond(W) f_2^\diamond (W) \cdots f_k^\diamond (W)$, where $f_j^\dagger(W)=(f_j(W))^\dagger$. The present paper studies sufficient conditions on $f_j$ to ensure such fractional calculus are unbounded normal operators. The results will be extended beyond continuous functions.

Abstract Let $x, y\in \mathbb{R}^n.$ We use the notation $x\prec_w y$ when $x$ is weakly majorized by $y$. We say that $x\prec_w y$ is decomposable at $k$ $(1\leq k < n)$ if $x\prec_w y$ has a coincidence at $k$ and $y_{k}\neq y_{k+1}$. Corresponding to this majorization we have a doubly substochastic matrix $P$. The paper presents $x\prec_w y$ is decomposable at some $k$ $(1\leq k<n)$ if and only if $P$ is of the form $D\oplus Q$ where $D$ and $Q$ are doubly stochastic and doubly substochastic matrices, respectively. Also, we write some algorithms to obtain $x$ from $y$ when $x\prec_w y$.

Abstract     In this paper we introduce approximate $\phi$-biprojective Banach algebras, where $\phi$ is a non-zero character. We show that for SIN group $G$, the group algebra $L^{1}(G)$ is approximately $\phi$-biprojective if and only if $G$ is amenable, where $\phi$ is the augmentation character. Also we show that the Fourier algebra $A(G)$ over a locally compact $G$ is always approximately $\phi$-biprojective.

Abstract In this paper, we study the K-Riesz bases and the K-g-Riesz bases in Hilbert spaces. We show that for $K \in B(\mathcal{H})$, a K-Riesz basis is precisely the image of an orthonormal basis under a bounded left-invertible operator such that the range of this operator includes the range of $K$. Also, we show that $\lbrace \Lambda_i \in B(\mathcal{H}, \mathcal{H}_i ) : \, i \in I \rbrace$ is a K-g-Riesz basis for $\mathcal{H}$ with respect to $\lbrace \mathcal{H}_i \rbrace_{i \in I}$
if and only if there exists a g-orthonormal basis $\lbrace Q_i \rbrace_{i \in I}$
for $\mathcal{H}$ and a bounded right-invertible operator $U $ on $\mathcal{H}$
such that $\Lambda_i = Q_i U$ for all $i \in I$, and $R(K) \subset R(U^{*})$.

Abstract In this paper, we generalize the Jensen's inequality for $m$-convex functions and we present a correction of Jensen's inequality which is a better than the generalization of this inequality for $m$-convex functions. ّFinally we have found new lower and upper bounds for Jensen's discrete inequality.

Abstract     In this paper, we first investigate the continuity of the spectral radius functions on continuous inverse algebras. Then we support our results by some examples. Finally, we continue our investigation by further determining the automatic continuity of linear mappings and homomorphisms on these algebras.

Abstract Let $\mathcal{A}$ be a Banach algebra with a left approximate identity.
    In this paper, under each of the following conditions, we prove that $\mathcal{A}$ is zero product determined.
    
    (i) For every continuous bilinear mapping $\phi$ from ${\mathcal A}\times {\mathcal A}$ into ${\mathcal X}$, where ${\mathcal X}$ is a Banach space, there exists $k>0$ such that 
    $\Vert \phi(a,b)\Vert\leq k \Vert ab\Vert$, for all $a,b\in\mathcal{A}$.
    
    (ii) $\mathcal{A}$ is generated by idempotents.