There is no difference with version 2, only an author whose name was forgotten in version 2 is added. arXiv admin note: text overlap with arXiv:2401.01390, arXiv:2401.01061
To complete the study of Fredholm type operators of [10] and [11], we define in this paper the classes of left and right semi-B-Fredholm operators (Definition 3.1). Then, we prove that an operator $T \in L(X), X$ being a Banach space, is a left( resp. right) semi-B-Fredholm operator if and only if $T$ is the direct sum of left(resp. right) semi-Fredholm operator and a nilpotent one. This result extend the earlier characterization of B-Fredholm operators as the direct sum of a Fredholm operator and a nilpotent one obtained in [4,Theorem 2.7].
Let $X$ a Banach space and $T$ a bounded linear operator on $X.$ We denote by $S(T)$ the set of all $λ\in \cit$ such that $T$ does not have the single-valued extension property at $λ$. In this note we prove equality up to $S(T)$ between the left Drazin spectrum and the left B-Fredholm spectrum and between the semi-essential approximate point spectrum and the left Drazin spectrum. As applications we investigate generalized Weyl's theorem for operator matrices and multipliers operators.
For bounded right linear operators, in a right quaternionic Hilbert space with a left multiplication defined on it, we study the approximate $S$-point spectrum. In the same Hilbert space, then we study the Fredholm operators and the Fredholm index. In particular, we prove the invariance of the Fredholm index under small norm operator and compact operator perturbations. Finally, in association with Fredholm operators, we develop the theory of essential S-spectrum. We also characterize the $S$-spectrum in terms of the essential S-spectrum and Fredholm operators. In the sequel we study left and r…
As we knew, study the perturbation theory of spectra of operator is a very important project in mathematics physics, in particular, in quantum mechanics. In this paper, we characterize the Fredholm perturbation for the Weyl spectrum, essential spectrum, spectrum, left spectrum, right spectrum, lower semi-Fredholm spectrum, upper semi-Weyl spectrum and lower semi-Weyl spectrum of upper triangular operator matrix $M_{C}=({cc} A&C 0&B)$.
Burgos, Kaidi, Mbekhta and Oudghiri provided an affirmative answer to a question of Kaashoek and Lay and proved that an operator $F$ is power finite rank if and only if $σ_{dsc}(T+F) =σ_{dsc}(T)$ for every operator $T$ commuting with $F$. Later, several authors extended this result to the essential descent spectrum, the left Drazin spectrum and the left essentially Drazin spectrum. In this paper, using the theory of operator with eventual topological uniform descent and the technique used in Burgos, Kaidi, Mbekhta, and Oudghiri, we generalize this result to various spectra originated from seni…
Muraleetharan and Thirulogasanthan in (J. Math. phys. 59, No. 10, 103506, 27p. (2018)) introduced the concept of Calkin Sspectrum of a bounded quaternionic linear operators. The study of this spectrum is establisched using the Fredholm operators theory. Motivated by this, we study the general framework of the Fredholm element with respect to a quaternionic Banach algebra homomorphism. First, we investigate the Fredholm S-spectrum of the sum of two elements in quaternionic Banach algebra by means of the Fredholm S-spectrum of the two elements. Next, we prove a perturbation result on this spectr…
We prove novel results on interpolation of Fredholm operators including an abstract factorization theorem. The main result of this paper provides sufficient conditions on the parameters $θ\in (0,1)$ and $q\in \lbrack 1,\infty ]$ under which an operator $A$ is a Fredholm operator from the real interpolation space $(X_{0},X_{1})_{θ,q}$ to $(Y_{0},Y_{1})_{θ,q} $ for a given operator $A\colon (X_{0},X_{1})\rightarrow (Y_{0},Y_{1})$ between compatible pairs of Banach spaces such that its restrictions to the endpoint spaces are Fredholm operators. These conditions are expressed in terms of the corre…
In this work, we improve and refine some numerical radius inequalities. In particular, for all Hilbert space operators $T$, the celebrated Kittaneh inequality reads: \begin{align*} \frac{1}{4}\left\| T^*T + TT^*\right\|\le w^{2 }\left(T \right) \le \frac{1}{2}\left\| T^*T + TT^*\right\|. \end{align*} In this work we provide some important refinements for the upper bound of the Kittaned inequality. Indeed, we establish \begin{align*} w^{2 }\left(T \right) \le \frac{1}{2}\left\| T^*T + TT^*\right\| - \frac{1}{4} \mathop {\inf }\limits_{\left\| x \right\| = 1} \left( {\left\langle {\left| T \righ…
Building upon the linear version of mixed summable sequences in arbitrary Banach spaces of A. Pietsch, we introduce a nonlinear version of his concept and study its properties. Extending previous work of J. D. Farmer, W. B. Johnson and J. A. Chávez-Domínguez, we define Lipschitz $\left(\mathfrak{m}^L\left(s;q\right),p\right)$ and Lipschitz $\left(p,\mathfrak{m}^L\left(s;q\right)\right)-$summing maps and establish inclusion theorems, composition theorems and several characterizations. Furthermore, we prove that the classes of Lipschitz $\left(r,\mathfrak{m}^L\left(r;r\right)\right)-$summing map…
New upper and lower bounds for the numerical radii of Hilbert space operators are given. Among our results, we prove that if $A\in \mathcal{B} \left( \mathcal{H}\right) $ is a hyponormal operator, then for all non-negative non-decreasing operator convex $f$ on $ [0,\infty ),$ we have \[f\left( ω\left( A \right) \right)\le \frac{1}{2}\left\| f\left( \frac{1}{1+\frac{ξ_{\left| A \right|}^{2}}{8}}\left| A \right| \right)+f\left( \frac{1}{1+\frac{ξ_{\left| A \right|}^{2}}{8}}\left| {{A}^{*}} \right| \right) \right\|,\] where ${{ξ}_{\left| A\right| }}=\underset{\left| x\right| =1}{\mathop{\inf }}\,…
We prove several numerical radius inequalities for linear operators in Hilbert spaces. It is shown, among other inequalities, that if $A$ is a bounded linear operator on a complex Hilbert space, then \[ω\left( A \right)\le \frac{1}{2}\sqrt{\left\| {{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}} \right\|+\left\| \left| A \right|\left| {{A}^{*}} \right|+\left| {{A}^{*}} \right|\left| A \right| \right\|},\] where $ω\left( A \right)$, $\left\| A \right\|$, and $\left| A \right|$ are the numerical radius, the usual operator norm, and the absolute value of $A$, respectively. This inequalit…
Let $α,β$ be orientation-preserving homeomorphisms of $[0,\infty]$ onto itself, which have only two fixed points at $0$ and $\infty$, and whose restrictions to $\mathbb{R}_+=(0,\infty)$ are diffeomorphisms, and let $U_α,U_β$ be the corresponding isometric shift operators on the space $L^p(\mathbb{R}_+)$ given by $U_μf=(μ')^{1/p}(f\circμ)$ for $μ\in\{α,β\}$. We prove sufficient conditions for the right and left Fredholmness on $L^p(\mathbb{R}_+)$ of singular integral operators of the form $A_+P_γ^++A_-P_γ^-$, where $P_γ^\pm=(I\pm S_γ)/2$, $S_γ$ is a weighted Cauchy singular integral operator, $…
In this work, a refinement of the Cauchy--Schwarz inequality in inner product space is proved. A more general refinement of the Kato's inequality or the so called mixed Schwarz inequality is established. Refinements of some famous numerical radius inequalities are also pointed out. As shown in this work, these refinements generalize and refine some recent and old results obtained in literature. Among others, it is proved that if $T\in\mathscr{B}\left(\mathscr{H}\right)$, then \begin{align*} ω^{2}\left(T\right) &\le \frac{1}{12} \left\| \left| T \right|+\left| {T^* } \right|\right\|^2 + \frac{1…
Let $A_i$ and $B_i$ be positive definite matrices for all $i=1,\cdots,m.$ It is shown that $$\left|\left|\sum_{i=1}^m(A_i^2\sharp B_i^2)^r\right|\right|_1\leq\left|\left|\left(\left(\sum_{i=1}^mA_i\right)^{\frac{pr}{_2}}\left(\sum_{i=1}^mB_i\right)^{pr}\left(\sum_{i=1}^mA_i\right)^{\frac{rp}{_2}}\right)^{\frac{1}{p}}\right|\right|_1,$$for all $p>0$ and for all $r\geq1.$ We conjecture this inequality is also true for all unitarily invariant norms. We give an affirmative answer to the case of $m=2,$ $p\geq1$, $r\geq1$ and for all unitarily invariant norms. In other words, it is shown that $$\lef…
Following an idea of Lin, we prove that if $A$ and $B$ be two positive operators such that $0<mI\le A\le m'I\le M'I\le B\le MI$, then \begin{equation*} {{Φ}^{2}}\left( \frac{A+B}{2} \right)\le \frac{{{K}^{2}}\left( h \right)}{{{\left( 1+\frac{{{\left( \log \frac{M'}{m'} \right)}^{2}}}{8} \right)}^{2}}}{{Φ}^{2}}\left( A\#B \right), \end{equation*} and \begin{equation*} {{Φ}^{2}}\left( \frac{A+B}{2} \right)\le \frac{{{K}^{2}}\left( h \right)}{{{\left( 1+\frac{{{\left( \log \frac{M'}{m'} \right)}^{2}}}{8} \right)}^{2}}}{{\left( Φ\left( A \right)\#Φ\left( B \right) \right)}^{2}}, \end{equation*} w…
Using the properties of geometric mean, we shall show for any $0\le α,β\le 1$, \[f\left( A{{\nabla }_{α}}B \right)\le f\left( \left( A{{\nabla }_{α}}B \right){{\nabla }_{β}}A \right){{\sharp}_{α}}f\left( \left( A{{\nabla }_{α}}B \right){{\nabla }_{β}}B \right)\le f\left( A \right){{\sharp}_{α}}f\left( B \right)\] whenever $f$ is a non-negative operator log-convex function, $A,B\in \mathcal{B}\left( \mathcal{H} \right)$ are positive operators, and $0\le α,β\le 1$. As an application of this operator mean inequality, we present several refinements of the Aujla subadditive inequality for operator …
We improve and generalize some operator inequalities for positive linear maps. It is shown, among other inequalities, that if $0<m\le B\le m'<M'\le A\le M$ or $0<m\le A\le m'<M'\le B\le M$, then for each $2\le p<\infty $ and $ν\in \left[ 0,1 \right]$, \begin{equation*} {{Φ}^{p}}\left( A{{\nabla }_{ν}}B \right)\le {{\left( \frac{K\left( h \right)}{{{4}^{\frac{2}{p}-1}}{{K}^{r}}\left( h' \right)} \right)}^{p}}{{Φ}^{p}}\left( A{{\#}_{ν}}B \right), \end{equation*} and \begin{equation*} {{Φ}^{p}}\left( A{{\nabla }_{ν}}B \right)\le {{\left( \frac{K\left( h \right)}{{{4}^{\frac{2}{p}-1}}{{K}^{r}}\lef…
In this article, we present some new general forms of numerical radius inequalities for Hilbert space operators. The significance of these inequalities follow from the way they extend and refine some known results in this field. Among other inequalities, it is shown that if $A$ is a bounded linear operator on a complex Hilbert space, then \[{{w}^{2}}\left( A \right)\le \left\| \int_{0}^{1}{{{\left( t\left| A \right|+\left( 1-t \right)\left| {{A}^{*}} \right| \right)}^{2}}dt} \right\|\le \frac{1}{2}\left\| \;{{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}} \right\|\] where $w\left( A \…
In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen's type and Jensen-Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen's operator inequality for strongly convex functions. As a corollary, we improve Hölder-McCarthy inequality under suitable conditions. More precisely we show that if $Sp\left( A \right)\subset I\subseteq \left( 1,\infty \right)$, then \[{{\left\langle Ax,x \right\rangle }^{r}}\le \left\langle {{A}^{r}}x,x \right\rangle -\frac{{{r}^{2}}-r}{2}\left( \left\langle {{A}^…
The main purpose of this paper is to introduce lacunary strong geometric zweier convergent sequence spaces $N_{θ}^{0} \left[Z\left(G\right)\right]$, $N_{θ} \left[Z\left(G\right)\right]$, $N_{θ}^{\infty } \left[Z\left(G\right)\right]$consisting of all sequences $x=\left(x_{k} \right)$such that $\left[Z\left(G\right)\right]x$ are in the spaces $N_{θ}^{0} ,N_{θ} {\rm and\; }N_{θ}^{\infty } $ respectively, which are normed. We prove certain topological properties of these spaces and compute their lacunary stastical zweier convergence.
In this article, we present some new inequalities for numerical radius of Hilbert space operators via convex functions. Our results generalize and improve earlier results by El-Haddad and Kittaneh. Among several results, we show that if $A\in \mathbb{B}\left( \mathcal{H} \right)$ and $r\ge 2$, then \[{{w}^{r}}\left( A \right)\le {{\left\| A \right\|}^{r}}-\underset{\left\| x \right\|=1}{\mathop{\inf }}\,{{\left\| {{\left| \left| A \right|-w\left( A \right) \right|}^{\frac{r}{2}}}x \right\|}^{2}}\] where $w\left( \cdot \right)$ and $\left\| \cdot \right\|$ denote the numerical radius and usual …
The purpose of this paper is to present some general inequalities for operator concave functions which include some known inequalities as a particular case. Among other things, we prove that if $A\in \mathcal{B}\left( \mathcal{H} \right)$ is a positive operator such that $mI\le A\le MI$ for some scalars $0<m<M$ and $Φ$ is a normalized positive linear map on $\mathcal{B}\left( \mathcal{H} \right)$, then \[\begin{aligned} {{\left( \frac{M+m}{2\sqrt{Mm}} \right)}^{r}}&\ge {{\left( \frac{\frac{1}{\sqrt{Mm}}Φ\left( A \right)+\sqrt{Mm}Φ\left( {{A}^{-1}} \right)}{2} \right)}^{r}} & \ge \frac{\frac{1}…
For a real normed space $X$, we study the $n$-dual space of $\left(X,\left\Vert \cdot \right\Vert \right) $ and show that the space is a Banach space. Meanwhile, for a real normed space $X$ of dimension $d\geq n$ which satisfies property (G), we discuss the $n$-dual space of $\left(X,\left\Vert \cdot,\ldots,\cdot \right\Vert _{G}\right) $, where $% \left\Vert \cdot,\ldots,\cdot \right\Vert _{G}$ is the Gähler $n$% -norm. We then investigate the relationship between the $n$-dual space of $% \left(X,\left\Vert \cdot \right\Vert \right) $ and the $n$-dual space of $% \left(X,\left\Vert \cdot,\ldo…
We present conditions for a family $\left(A\left(x\right)\right)_{x\in\mathbb{R}^{d}}$ of self-adjoint operators in $H^{r}=\mathbb{C}^{r}\otimes H$ for a separable complex Hilbert space $H$, such that the Callias operator $D=ic\nabla+A\left(X\right)$ satisfies that $\left(D^{\ast}D+1\right)^{-N}-\left(DD^{\ast}+1\right)^{-N}$ is trace class in $L^2\left(\mathbb{R}^{d},H^{r}\right)$. Here, $c\nabla$ is the Dirac operator associated to a Clifford multiplication $c$ of rank $r$ on $\mathbb{R}^{d}$, and $A\left(X\right)$ is fibre-wise multiplication with $A\left(x\right)$ in $L^2\left(\mathbb{R}^{…
In this paper we consider the problem of characterizing linear maps on special $ \star $-algebras behaving like left or right centralizers at orthogonal elements and obtain some results in this regard.
In this paper, we present several sharp upper bounds for the numerical radii of the diagonal and off-diagonal parts of the $2\times2$ block operator matrix $\begin{bmatrix}A&B\\ C&D\end{bmatrix}$. Among extensions of some results of Kittaneh et al., it is shown that if $T=\begin{bmatrix}A&0\\ 0&D\end{bmatrix}$, and $f$ and $g$ are non-negative continuous functions on $[0,\infty)$ such that $f(t)g(t)=t\,\,(t\geq 0)$, then for all nonnegative nondecreasing convex functions $h$ on $[0,\infty)$ , we obtain that \begin{align*}h\left(w^r(T)\right)\leq \max\left(\left\|\frac{1}{p}h\left(f^{pr}(\left|…
In this work, the mixed Schwarz inequality for semi-Hilbertian space operators is proved. Namely, for every positive Hilbert space operator $A$. If $f$ and $g$ are nonnegative continuous functions on $\left[0,\infty\right)$ satisfying $f(t)g(t) =t$ $(t\ge0)$, then \begin{align*} \left| {\left\langle {T x,y} \right\rangle_A } \right| \le \left\| {f\left( {\left| T \right|_A x} \right)} \right\|_A \left\| {g\left( {\left| {T^{\sharp_A } } \right|_A y} \right)} \right\|_A \end{align*} for every Hilbert space operator $T$ such that the range of $T^* A$ is a subset in the range of $A$, such that $A…
Given an arbitrarily weak notion of left-<f>-porosity and an arbitrarily strong notion of right-<g>-porosity, we construct an example of closed subset of the real line which is not sigma-left-<f>-porous and is right-<g>-porous. We also briefly summarize the relations between three different definitions of porosity controlled by a function; we then observe that our construction gives the example for any combination of these definitions of left-porosity and right-porosity.
In this paper we obtain the sharp estimates for the mixed $\left( \ell_{1},\ell _{2}\right) $-Littlewood inequality for real scalars with exponents $\left(2,1,2,2....,2\right) .$ These results are applied to find sharp estimates for the constants of a family of $3$-linear Bohnenblust--Hille inequalities with multiple exponents.
Let $L$ be a self-adjoint invertible operator in a Hilbert space such that $L^{-1}$ is $p$-summable. Under a certain discrete dimension spectrum assumption on $L$, we study the relation between the (regularized) Fredholm determinant, $\det_{p}(I+z\cdot L^{-1})$, on the one hand and the zeta regularized determinant, $\det_ζ(L+z)$, on the other. One of the main results is the formula \begin{equation*} \frac{\det_ζ (L + z)}{\det_ζ (L)} = \exp \left( \sum_{j=1}^{p-1} \frac{z^j}{j!} \cdot \frac{d^j}{dz^j} \log \det\nolimits_ζ (L+z) |_{z=0} \right) \cdot \det\nolimits_{p}(I + z \cdot L^{-1} ). \end{…
We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only …
Present work contains a method to obtain Jackson and Stechkin type inequalities of approximation by integral functions of finite degree (IFFD) in some variable exponent Lebesgue space of real functions defined on $\boldsymbol{R}:=\left( -\infty ,+\infty \right) $. To do this we employ a transference theorem which produce norm inequalities starting from norm inequalities in $\mathcal{C}(\boldsymbol{R})$, the class of bounded uniformly continuous functions defined on $\boldsymbol{R}$. Let $B\subseteq \boldsymbol{R}$ be a measurable set, $p\left( x\right) :B\rightarrow \lbrack 1,\infty )$ be a me…
We analyze the role played by $n$-convexity for the fulfillment of a series of linear functional inequalities that extend the Hornich-Hlawka functional inequality, $f\left( x\right) +f\left( y\right) +f\left( z\right) +f\left( x+y+z\right) \geq f\left( x+y\right) +f\left( y+z\right)+f\left( z+x\right) +f(0),$ including extensions to the case of positive operators.
To extend the Euclidean operator radius, we define $w_p$ for an $n$-tuples of operators $(T_1,\ldots, T_n)$ in $\mathbb{B}(\mathscr{H})$ by $w_p(T_1,\ldots,T_n):= \sup_{\| x \| =1} \left(\sum_{i=1}^{n}| \langle T_i x, x \rangle |^p \right)^{\frac1p}$ for $p\geq1$. We generalize some inequalities including Euclidean operator radius of two operators to those involving $w_p$. Further we obtain some lower and upper bounds for $w_p$. Our main result states that if $f$ and $g$ are nonnegative continuous functions on $\left[ 0,\infty \right) $ satisfying $f\left( t\right) g\left(t\right) =t$ for all …
Suppose $X$ is a compact Hausdorff space. In terms of topolocical properties of $X$, we find topological conditions on $X$ that are equivalent to each of the following: 1. every additive local multiplication on $C\left( X\right) $ is a multiplication, 2. every additive local multiplication on $C_{R}\left( X\right) $ is a multiplication, and 3. every additive map on $C\left( X\right) $ that is zero-preserving (i.e., $f\left( x\right) =0$ implies $\left( Tf\right) \left( x\right) =0$) has the form $T\left( f\right) =T\left( 1\right) \operatorname{Re}f+T\left( i\right) \operatorname{Im}f$.
