Sentence examples for function is operator from inspiring English sources

The log function is (operator) Gâteaux differentiable with the following explicit formula for the derivative (cf. Pedersen [27, p.155]): bigl nablalog(A bigr) (B =int_{0}^{infty} (sI+A)^{-1}B sI+A)^{-1}B sI+Ar (A,Binmathcal{A}(H)) and I the identity operator.

A function f is called operator concave if the function −f is operator convex.

A real-valued continuous function f on an interval I is said to be operator convex (operator concave) if f ( ( 1 − λ ) A + λ B ) ≤ ( 1 − λ ) f ( A ) + λ f ( B ). in the operator order for all λ ∈ [ 0, 1 ] and for every self-adjoint operator A and B on a Hilbert space H whose spectra are contained in I. Notice that a function f is operator concave if −f is operator convex.

If the function f is operator convex, then the so-called Jensen operator inequality (f(Phi(X))leqPhi(f(X))) holds for any unital positive linear mapping Φ on (mathcal{B}(H)) and any (X inmathcal{B}_{h}(H)) with spectrum contained in I.

Clearly every nonnegative operator convex function is of operator P-class.

We show that every nonnegative operator convex function is of operator P-class, but the converse is not true in general.

The function tα (0<α<1) is operator monotone on 0⩽t<∞.

First of all, if f is an operator Lipschitz function on ({mathbb R}), then f is differentiable everywhere on ({mathbb R}) which was established in [34] (but not necessarily continuously differentiable: the function (xmapsto x^2sin (1/x)) is operator Lipschitz, see [38]).

Echocardiography is the most commonly used imaging modality to assess the severity of VHD and left ventricular (LV) function; however it is operator dependent and has several limitations.

The detector function is an operator that transforms the image locally, and often involves spatial derivatives of the image.

Since (1leq1+rleq2), the function (trightarrow t^{1+r}) is operator convex.