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For a tuple of operators $\boldsymbol{A}= (A_1, \ldots, A_d)$, $\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})$ is defined as $\min\limits_{\boldsymbol{z} \in \mathbb C^d} \|\boldsymbol{A-zI}\|$ and $\text{var}_x (\boldsymbol{A})$ as $\|\boldsymbol{A} x\|^2-\sum_{j=1}^d {\big|}\langle x| A_j x\rangle{\big|}^2.$ For a tuple $\boldsymbol{A}$ of commuting normal operators, it is known that $$\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})^2=\sup_{\|x\|=1}\text{var}_x (\boldsymbol{A}).$$ We give an expression for the maximal joint numerical range of a tuple of doubly commuting matrices. Consequently, we obtain that the above distance formula holds for tuples of doubly commuting matrices. We also discuss some general conditions on the tuples of operators for this formula to hold. As a result, we obtain that it holds for tuples of Toeplitz operators as well., Comment: to appear in LAA

The matrix $S = [1+x_i y_j]_{i,j=1}^{n}, 0

An expression for the subdifferential of the joint numerical radius is obtained. Its applications to the best approximation problems in the joint numerical radius are discussed.

For a unital $C^*$-algebra $\mathcal A$ and a subspace $\mathcal B$ of $\mathcal A$, a characterization for a best approximation to an element of $\mathcal A$ in $\mathcal B$ is obtained. As an application, a formula for the distance of an element of $\mathcal A$ from $\mathcal B$ has been obtained, when a best approximation of that element to $\mathcal B$ exists. Further, a characterization for Birkhoff-James orthogonality of an element of a Hilbert $C^*$-module to a subspace is obtained., Comment: To appear in Journal of the Ramanujan Mathematical Society

In the last few decades, the concept of Birkhoff-James orthogonality has been used in several applications. In this survey article, the results known on the necessary and sufficient conditions for Birkhoff-James orthogonality in certain Banach spaces are mentioned. Their applications in studying the geometry of normed spaces are given. The connections between this concept of orthogonality, and the Gateaux derivative and the subdifferential set of the norm function are provided. Several interesting distance formulas can be obtained using the characterizations of Birkhoff-James orthogonality, which are also mentioned. In the end, some new results are obtained., Comment: In Equation (4.1), the factor of 2 is misplaced and it should be on the left-hand side of the equation

It is shown that for positive real numbers $ 0<\lambda_{1}<\dots<\lambda_{n}$, $\left[\frac{1}{\beta({\lambda_i}, {\lambda_j})}\right]$, where $ \beta(\cdot,\cdot)$ denotes the beta function, is infinitely divisible and totally positive. For $ \left[\frac{1}{\beta({i},{j})}\right]$, the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let $\mathfrak w(n)$ be the $n$th Bell number. It is proved that $\left[\mathfrak w(i+j)\right]$ is a totally positive matrix but is infinitely divisible only upto order $4$. It is also shown that the symmetrized Stirling matrices are totally positive., Comment: 9 pages

We obtain a necessary and sufficient condition for a matrix $A$ to be Birkhoff-James orthogonal to any subspace $\mathscr W$ of $\mathbb M_n(\mathbb C)$. Using this we obtain an expression for the distance of $A$ from any unital $C^*$ subalgebra of $\mathbb M_n(\mathbb C)$., Comment: 12 pages

Perturbation bounds for Mostow's decomposition and the bipolar decomposition of matrices have been computed. To do so, expressions for the derivative of the geometric mean of two positive definite matrices have been derived., Comment: 16 pages, 2 figures, to appear in Linear Algebra and its Applications

We obtain necessary and sufficient conditions for a matrix $A$ to be Birkhoff-James orthogonal to another matrix $B$ in the Ky Fan $k$-norms. A characterization for $A$ to be Birkhoff-James orthogonal to any subspace $\mathscr W$ of $\mathbb M(n)$ is also obtained., Comment: 15 pages

Inequalities for norms of different versions of the geometric mean of two positive definite matrices are presented., Comment: 10 pages

The norm of the $m$th derivative of the map that takes an operator to its $k$th antisymmetric tensor power is evaluated. The case $m=1$ has been studied earlier by Bhatia and Friedland [R. Bhatia and S. Friedland, Variation of Grassman powers and spectra, Linear Algebra and its Applications, 40:1--18, 1981]. For this purpose a multilinear version of a theorem of Russo and Dye is proved: it is shown that a positive $m$-linear map between $C^{\ast}$-algebras attains its norm at the $m$-tuple $(I, \, I, ..., I).$ Expressions for derivatives of the maps that take an operator to its $k$th tensor power and $k$th symmetric tensor power are also obtained. The norms of these derivatives are computed. Derivatives of the map taking a matrix to its permanent are also evaluated., Comment: 15 pages

Perturbation or error bounds of functions have been of great interest for a long time. If the functions are differentiable, then the mean value theorem and Taylor's theorem come handy for this purpose. While the former is useful in estimating $\|f(A+X)-f(A)\|$ in terms of $\|X\|$ and requires the norms of the first derivative of the function, the latter is useful in computing higher order perturbation bounds and needs norms of the higher order derivatives of the function. In the study of matrices, determinant is an important function. Other scalar valued functions like eigenvalues and coefficients of characteristic polynomial are also well studied. Another interesting function of this category is the permanent, which is an analogue of the determinant in matrix theory. More generally, there are operator valued functions like tensor powers, antisymmetric tensor powers and symmetric tensor powers which have gained importance in the past. In this article, we give a survey of the recent work on the higher order derivatives of these functions and their norms. Using Taylor's theorem, higher order perturbation bounds are obtained. Some of these results are very recent and their detailed proofs will appear elsewhere., Comment: 17 pages