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Orthogonality and the numerical range

1975; Mathematical Society of Japan; Volume: 27; Issue: 3 Linguagem: Inglês

10.2969/jmsj/02730405

1881-1167

Mary R. Embry,

Statistical and numerical algorithms

In this paper we shall expand upon results and techniques developed in [2] to investigate certain geometric relationships between a complex Hilbert space $X$ and the numerical range of a continuous linear operator $A$ on $X$ .In Section 2 we present a version of the Cauchy-Schwartz inequality valid in the boundary of the numerical range of $A$ .In Section 3 we study the action on elements $z$ of $W(A)$ induced by the action of $A$ on elements $x$ of $X$ such that $\langle Ax, x\rangle/\Vert x\Vert^{2}=z$ .The numerical range of $t4$ is the set of complex numbers, $ W(A)=\{\langle Ax, x\rangle$ :$x\in X$ and $\Vert x\Vert=1$ }, where $\langle, \rangle$ is the given inner product on $X$ and $\Vert\Vert$ is the associated norm.Basic properties of the numerical range are discussed in [4].In particular the Hausdorff-Toeplitz theorem is proven: $W(A)$ is convex.We use the following terminology: $z$ is an extreme point of $W(A)$ if $z\in W(A)$ andis not in the interior of any line segment lying in $W(A);L$ is a line of suP- Port for $W(A)$ if $W(A)$ lies in one of the two closed half-planes determined by $L$ and $L$ contains at least one point of the closure of $W(A);b$ and $c$ are adjacent extreme pOints of $W(A)$ if the line segment joining $b$ and $c$ lies in the boundary of $W(A);c$ is a corner of $W(A)$ if $c$ is an extreme point of $W(A)$ and there exist more than one line of support for $W(A)$ passing through $c$ .We define the set $M_{z}$ for each complex $z$ by $M_{z}=\{x:x\in X$ and $\langle Ax, x\rangle$ $=z\Vert x\Vert^{2}\}$ .\S 2. A Cauchy-Schwartz inequality.Consider a line of support $L$ for $W(A)$ and the associated set in $X,$ $N=$ $\{x:\langle Ax, x\rangle=z\Vert x\Vert^{2}, z\in L\}$ .In [2] we proved that $N$ is a closed linear sub- space of $X$ and that $A$ behaves very much like an Hermitian operator on $N$ .More precisely LEMMA 2.1.Let $L$ be a line of $suPPort$ of $W(A)$ and $N=\{x;\langle Ax, x\rangle=$ $z\Vert x\Vert^{2},$ $z\in L$ }.Let $\theta=0$ if $L$ is horizontal; otherwise $\theta$ is the measure of the acute angle between $L$ and the x-axis.Then i) $N$ is a closed linear subspace of $X$ , and M. R.