Non Commutative Bernstein Inequality

Question

In the proof for noncommutative Bernstein Inequality, a symbol is used whose intuition is not clear.

In the picture, Pr[X symbol A] is used. What is the semantic meaning of that symbol.

Can someone explain the first line of proof of operator Markov inequality given in this picture.

Answer 1

Quoting Wikipedia:

For arbitrary square matrices $M, N$ we write $M\succeq N$ if $M - N \succeq 0$; i.e., $M - N$ is positive semi-definite. This defines a partial ordering on the set of all square matrices. One can similarly define a strict partial ordering $M \succ N$.

Your symbol is the negation of this. I.e., $M\not\succeq N$ if $M - N$ is not positive semi-definite.

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Regarding the first sentence of the proof. $X \not\preceq A$ is equivalent to $A-X$ not positive semi-definite, and therefore equivalent to $A^{-1/2}(A-X)A^{1/2} = I-A^{-1/2}X A^{1/2}$ not positive semi-definite. I.e., with the notations, $A^{-1/2}X A^{1/2} \not\preceq I$. What does that mean? It means the at least one of the eigenvalues of $I-A^{-1/2}X A^{1/2}$ is negative, or (again, equivalently) that at least one of the eigenvalues of $A^{-1/2}X A^{1/2}$ is $> 1$. This is precisely saying $\lVert A^{-1/2}X A^{1/2}\rVert > 1$. (The spectral norm being the (absolute value of the) maximum eigenvalue.)