arXiv subjects
Publications and source records attributed to Kallol Paul.
We develope new lower bounds for the $A$-numerical radius of semi-Hilbertian space operators, and applying these bounds we obtain upper bounds for the $A$-numerical radius of the commutators of operators. The bounds obtained here improve on the existing ones. Further, we provide characterizations for the equality of the existing $A$-numerical radius inequalities of semi-Hilbertian space operators.
We present some new upper and lower bounds for the numerical radius of bounded linear operators on a complex Hilbert space and show that these are stronger than the existing ones. In particular, we prove that if $A$ is a bounded linear operator on a complex Hilbert space $\mathcal{H}$ and if $\Re(A)$, $\Im(A)$ are the real part, the imaginary part of $A$, respectively, then $$ w(A)\geq\frac{\|A\|}{2} +\frac{1}{2\sqrt{2}} \Big | \|\Re(A)+\Im(A)\|-\|\Re(A)-\Im(A)\| \Big | $$ and $$ w^2(A)\geq\frac{1}{4}\|A^*A+AA^*\|+\frac{1}{4}\Big| \|\Re(A)+\Im(A)\|^2-\|\Re(A)-\Im(A)\|^2\Big|. $$ Here $w(.)$ and $\|.\|$ denote the numerical radius and the operator norm, respectively. Further, we obtain refinement of inequalities for the numerical radius of the product of two operators. Finally, as an application of the second inequality mentioned above, we obtain an improvement of upper bound for the numerical radius of the commutators of operators.
Several upper and lower bounds for the numerical radius of $2 \times 2$ operator matrices are developed which refine and generalize the earlier related bounds. In particular, we show that if $B,C$ are bounded linear operators on a complex Hilbert space, then \begin{eqnarray*} && \frac{1}{2}\max \left \{ \|B\|, \|C\| \right \}+\frac{1}{4} \left | \|B+C^*\|-\|B-C^*\| \right | &&\leq w \left(\left[\begin{array}{cc} 0 & B C& 0 \end{array}\right]\right)\\ &&\leq \frac{1}{2} \max \left\{\|B\|,\|C\|\right \}+\frac{1}{2}\max \left \{r^{\frac{1}{2}}(|B||C^*|),r^{\frac{1}{2}}(|B^*||C|)\right\}, \end{eqnarray*} where $w(.)$, $r(.)$ and $\|.\|$ are the numerical radius, spectral radius and operator norm of a bounded linear operator, respectively. We also obtain equality conditions for the numerical radius of the operator matrix $\left[\begin{array}{cc} 0 & B C& 0 \end{array}\right]$. As application of results obtained, we show that if $B,C$ are self-adjoint operators then, $\max \Big \{\|B+C\|^2 , \|B-C\|^2 \Big\}\leq \left \|B^2+C^2 \right \|+2w(|B||C|). $
Let $A$ be a bounded linear operator on a complex Hilbert space and $\Re(A)$ ( $\Im(A)$ ) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of $A$, we prove that \begin{eqnarray*} w(A)&\geq &\frac{1}{2} \left \|A \right\| + \frac{ 1}{2} \mid \|\Re(A)\|-\|\Im(A)\|\mid,\,\,\mbox{and}\\ w^2(A)&\geq& \frac{1}{4} \left \|A^*A+AA^* \right\| + \frac{1}{2}\mid \|\Re(A)\|^2-\|\Im(A)\|^2 \mid, \end{eqnarray*} where $w(A)$ is the numerical radius of the operator $A$. We study the equality conditions for $w(A)=\frac{1}{2}\sqrt{\|A^*A+AA^*\|}$ and prove that $w(A)=\frac{1}{2}\sqrt{\|A^*A+AA^*\|} $ if and only if the numerical range of $A$ is a circular disk with center at the origin and radius $\frac{1}{2}\sqrt{\|A^*A+AA^*\|} $. We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones.
The purpose of the article is to generalize the concept of approximate Birkhoff-James orthogonality, in the semi-Hilbertian structure. Given a positive operator $ A $ on a Hilbert space $ \mathbb{H}, $ we define $ (ε,A)- $approximate orthogonality and $ (ε,A)- $approximate orthogonality in the sense of Chmieli$\acute{n}$ski and establish a relation between them. We also characterize $ (ε,A)- $approximate orthogonality in the sense of Chmieli$\acute{n}$ski for $A$-bounded and $A$-bounded compact operators. We further generalize the concept of right symmetric and left symmetric operators on a Hilbert space. The utility of these notions are illustrated by extending some of the previous results obtained by various authors in the setting of Hilbert spaces.
If $A,B$ are bounded linear operators on a complex Hilbert space, then % $w(A) \leq \frac{1}{2}\left( \|A\|+\sqrt{r\left(|A||A^*|\right)}\right)$ and $w(AB \pm BA)\leq 2\sqrt{2}\|B\|\sqrt{ w^2(A)-\frac{c^2(\Re (A))+c^2(\Im (A))}{2} },$ \begin{eqnarray*} w(A) &\leq& \frac{1}{2}\left( \|A\|+\sqrt{r\left(|A||A^*|\right)}\right),\\ w(AB \pm BA)&\leq& 2\sqrt{2}\|B\|\sqrt{ w^2(A)-\frac{c^2(\Re (A))+c^2(\Im (A))}{2} }, \end{eqnarray*} where $w(.),\|.\|,c(.)$ and $r(.)$ are the numerical radius, the operator norm, the Crawford number and the spectral radius respectively, and $\Re (A)$, $\Im (A)$ are the real part, the imaginary part of $A$ respectively. The inequalities obtained here generalize and improve on the existing well known inequalities.
New inequalities for the $A$-numerical radius of the products and sums of operators acting on a semi-Hilbert space, i.e. a space generated by a positive semidefinite operator $A$, are established. In particular, it is proved for operators $T$ and $S,$ having $A$-adjoint, that $$ ω_A(TS) \leq \frac{1}{2}ω_A(ST)+\frac{1}{4}\Big(\|T\|_A\|S\|_A+\|TS\|_A\Big),$$ where $ω_A(T)$ and $\|T\|_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$.
We characterize $k-$smoothness of an element on the unit sphere of a finite-dimensional polyhedral Banach space. Then we study $k-$smoothness of an operator $T \in \mathbb{L}(\ell_{\infty}^n,\mathbb{Y}),$ where $\mathbb{Y}$ is a two-dimensional Banach space with the additional condition that $T$ attains norm at each extreme point of $B_{\ell_{\infty}^{n}}.$ We also characterize $k-$smoothness of an operator defined between $\ell_{\infty}^3$ and $\ell_{1}^3.$
We give new inequalities for $A$-operator seminorm and $A$-numerical radius of semi-Hilbertian space operators and show that the inequalities obtained here generalize and improve on the existing ones. Considering a complex Hilbert space $\mathcal{H}$ and a non-zero positive bounded linear operator $A$ on $\mathcal{H},$ we show with among other seminorm inequalities, if $S,T,X\in \mathcal{B}_A(\mathcal{H})$, i.e., if $A$-adjoint of $S,T,X$ exist then $$2\|S^{\sharp_A}XT\|_A \leq \|SS^{\sharp_A}X+XTT^{\sharp_A}\|_A.$$ Further, we prove that if $T\in \mathcal{B}_A(\mathcal{H})$ then \begin{eqnarray*} \frac{1}{4}\|T^{\sharp_{A}}T+TT^{\sharp_{A}}\|_A \leq \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg), ~~\textit{and} \end{eqnarray*} \begin{eqnarray*} \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg) +\frac{1}{8}c_A^2\big(T+T^{\sharp_{A}}\big)+\frac{1}{8}c_A^2\big(T-T^{\sharp_{A}}\big) \leq w^2_A(T). \end{eqnarray*} Here $w_A(.), c_A(.)$ and $\|.\|_A $ denote $A$-numerical radius, $A$-Crawford number and $A$-operator seminorm, respectively.
Several refinements of norm and numerical radius inequalities of bounded linear operators on a complex Hilbert space are given. In particular, we show that if $A$ is a bounded linear operator on a complex Hilbert space, then $$ \frac{1}{4}\|A^*A+AA^*\| \leq \frac{1}{8}\bigg( \|A+A^*\|^2+\|A-A^*\|^2 +c^2(A+A^*)+c^2(A-A^*)\bigg) \leq w^2(A)$$ and \begin{eqnarray*} \frac{1}{2}\|A^*A+AA^*\| - \frac{1}{4}\bigg\|(A+A^*)^2 (A-A^*)^2 \bigg\|^{1/2} \leq w^2(A) \leq \frac{1}{2}\|A^*A+AA^*\|, \end{eqnarray*} %$$ \frac{1}{4}\|A^*A+AA^*\| \leq \frac{1}{2}w^2(A) + \frac{1}{8}\bigg\|(A+A^*)^2 (A-A^*)^2 \bigg\|^{1/2}\leq w^2(A),$$ where $\|.\|$, $w(.)$ and $c(.)$ are the operator norm, the numerical radius and the Crawford number, respectively. Further, we prove that if $A,D$ are bounded linear operators on a complex Hilbert space, then \begin{eqnarray*} \|AD^*\| \leq \left\| \int_0^1 \left( (1-t) \left(\frac{ |A|^2+|D|^2}{2}\right) +t\|AD^*\|I \right)^2dt \right\|^{1/2} \leq \frac{1}{2}\left\| |A|^2+|D|^2 \right\|, \end{eqnarray*} where $|A|=(A^*A)^{1/2}$ and $|D|=(D^*D)^{1/2}$. This is a refinement of well known inequality obtained by Bhatia and Kittaneh.
We obtain bounds for the numerical radius of $2 \times 2$ operator matrices which improve on the existing bounds. We also show that the inequalities obtained here generalize the existing ones. As an application of the results obtained here we estimate the bounds for the zeros of a monic polynomial and illustrate with numerical examples that the bounds are better than the existing ones.
Let $\mathcal{H}$ be a complex Hilbert space and let $A$ be a positive operator on $\mathcal{H}$. We obtain new bounds for the $A$-numerical radius of operators in semi-Hilbertian space $\mathcal{B}_A(\mathcal{H})$ that generalize and improve on the existing ones. Further, we estimate bounds for the $B$-operator seminorm and $B$-numerical radius of $2\times 2$ operator matrices, where $B=\mbox{diag}(A,A)$. The bounds obtained here improve on the existing ones.
New inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space $\mathcal{H}$ are given. In particular, it is established that if $T$ is a bounded linear operator on a Hilbert space $\mathcal{H}$ then \[ w^2(T)\leq \min_{0\leq α\leq 1} \left \| αT^*T +(1-α)TT^* \right \|,\] where $w(T)$ is the numerical radius of $T.$ The inequalities obtained here are non-trivial improvement of the well-known numerical radius inequalities. As an application we estimate bounds for the zeros of a complex monic polynomial.
We obtain upper and lower bounds for the Davis-Wielandt radius of bounded linear operators defined on a complex Hilbert space, which improve on the existing ones. We also obtain bounds for the Davis-Wielandt radius of operator matrices. We determine the exact value of the Davis-Wielandt radius of two special type of operator matrices $\left(\begin{array}{cc} I & B 0 & 0 \end{array}\right)$ and $\left(\begin{array}{cc} 0 & A B & 0 \end{array}\right)$, where $A,B\in \mathcal{B}(\mathcal{H})$, $I$ and $0$ are the identity operator and the zero operator on $\mathcal{H},$ respectively. Finally we obtain bounds for the Davis-Wielandt radius of operator matrices of the form $\left(\begin{array}{cc} A& B 0 & C \end{array}\right),$ where $A,B, C\in \mathcal{B}(\mathcal{H}).$
We characterize $k-$smoothness of bounded linear operators defined between infinite-dimensional Hilbert spaces. We study the problem in the setting of both finite and infinite-dimensional Banach spaces. We also characterize $k-$smoothness of operators on some particular spaces, namely $\mathbb{L}(\mathbb{X},\ell_{\infty}^n),~\mathbb{L}(\ell_{\infty}^3,\mathbb{Y}),$ where $\mathbb{X}$ is a finite-dimensional Banach space and $\mathbb{Y}$ is a two-dimensional Banach space. As an application, we characterize extreme contractions on $\mathbb{L}(\ell_{\infty}^3,\mathbb{Y}),$ where $\mathbb{Y}$ is a two-dimensional polygonal Banach space.
We study the Davis-Wielandt shell and the Davis-Wielandt radius of an operator on a normed linear space $\mathcal{X}$. We show that after a suitable modification, the modified Davis-Wielandt radius defines a norm on $\mathcal{L}(\mathcal{X})$ which is equivalent to the usual operator norm on $\mathcal{L}(\mathcal{X})$. We introduce the Davis-Wielandt index of a normed linear space and compute its value explicitly in case of some particular polyhedral Banach spaces. We also present a general method to estimate the Davis-Wielandt index of any polyhedral Banach space.
We characrterize extreme contractions defined between \ finite-dimensional polyhedral Banach spaces using $k$- smoothness of operators. We also explore weak L-P property, a recently introduced concept in the study of extreme contractions. We obtain a sufficient condition for a pair of finite-dimensional polyhedral Banach spaces to satisfy weak L-P property. As an application of these results, we explicitly compute the number of extreme contractions in some special Banach spaces. Our approach in this paper in studying extreme contractions lead to the improvement and generalization of previously known results.
We explore the relation between the orthogonality of bounded linear operators in the space of operators and that of elements in the ground space. To be precise, we study if $ T, A \in \mathbb{L}(\mathbb{X}, \mathbb{Y}) $ satisfy $ T \bot_B A,$ then whether there exists $ x \in \mathbb{X} $ such that $ Tx\bot_B Ax$ with $ \|x\| =1, \|Tx\| = \|T\|$, where $\mathbb{X}, \mathbb{Y} $ are normed linear spaces. In this context, we introduce the notion of Property $ P_n $ for a Banach space and illustrate its connection with orthogonality of a bounded linear operator between Banach spaces. We further study Property $ P_n $ for various polyhedral Banach spaces.