Birkhoff james orthogonality and distance formulas in c Algebras and for tuples of operators

Abstract

In a given normed space V, an element v\in V is said to be Birkhoff-James orthogonal to a subspace W of V if \|v\|\leq \|v-w\| for all w\in W. Let \mathcal A be a C^*-algebra. We show that a\in \mathcal A is Birkhoff-James orthogonal to a subspace \mathcal B of \mathcal A if and only if there exists a state \phi on \mathcal A such that \phi(a^*a) = \|a\|^2 and \phi(a^*b) = 0 for all b\in \mathcal B. newlineSince \|\cdot\| is a convex function, \lim\limits_{t\rightarrow 0^+} \dfrac{\|v+\lambda w\|-\|v\|}{t} always exists. This is known as the Gateaux derivative of \|\cdot\| at v. We shall give an expression for the Gateaux derivative of the C* norm in terms of states on \mathcal A. This will also give us alternate proofs or generalizations of various known results on the related notions of subdifferential sets, smooth points and norm parallelism. We shall show how to extend our results when a belongs to an ideal of \mathcal A. We shall generalize some of our results to Hilbert C*-modules. We shall also provide applications of the characterization of Birkhoff-James orthogonality in C*-algebras to find some distance formulas in \mathcal A, which are dependent only on the algebraic structure of \mathcal A. newlineLet \mathcal H be a Hilbert space and \mathscr B(\mathcal H) be the space of bounded linear operators on \mathcal H. Let A_1,\dots, A_d\in\mathscr B(\mathcal H). Let \boldsymbol{A}=(A_1, \ldots, A_d), then dist(\boldsymbol{A}, \mathbb C^d \boldsymbol{I}) is defined as \min\limits_{\boldsymbol{z} \in \mathbb C^d} \|\boldsymbol{A-z I}\| and var_x (\boldsymbol{A}) is defined as \|\boldsymbol{A} x\|^2-\sum_{j=1}^d {\big|}\langle x| A_j x\rangle{\big|}^2. We prove that when A_1, \dots, A_d are doubly commuting matrices or A_1,\dots, A_d are Toeplitz operators, then dist \left( \boldsymbol{A}, \mathbb C^d \boldsymbol{I}\right)^2 = \sup_{\|x\|=1} var_x \left(\boldsymbol{A}\right).We also give some equivalent conditions for any tuple of operators on any Hilbert space to satisfy the above distance formula.