2.1 Comparison with multiples of the identity
Let \(\iota \) be a finite set; all matrices below are in \(\mathbb {C}^{\iota \times \iota }\).
(a) Let \(A\) be Hermitian and \(c\) a real number with \(\lambda _i(A) \leq c\) for every eigenvalue. Then \(A \preceq c \cdot 1\).
(b) Every positive semidefinite \(A\) satisfies \(A \preceq (\operatorname {Tr}A) \cdot 1\).
(c) If \(A\) and \(B\) are positive definite with \(A \preceq B\), then \(B^{-1} \preceq A^{-1}\).
(d) If \(A\) is positive definite and \(A^{-1} \preceq c \cdot 1\) for some \(c {\gt} 0\), then \(c^{-1} \cdot 1 \preceq A\).
(a) By the spectral theorem \(c \cdot 1 - A\) is unitarily conjugate to the diagonal matrix with entries \(c - \lambda _i(A) \geq 0\), and conjugation by a unitary preserves positive semidefiniteness.
(b) The trace is the sum of the eigenvalues of \(A\), all of which are nonnegative, so every single eigenvalue is at most \(\operatorname {Tr}A\), and (a) applies with \(c = \operatorname {Tr}A\).
(c), (d) Both are read off Mathlib’s antitonicity of the inverse in a C\(^*\)-algebra with a star ordering, applied to the units of \(\mathbb {C}^{\iota \times \iota }\) with the \(\ell _2\)-operator norm; the inverse of \(c \cdot 1\) is \(c^{-1} \cdot 1\).
Let \(A\) be a C\(^*\)-algebra with a star ordering, let \(B \in A\) be positive and invertible, let \(X \in A\) and let \(t {\gt} 0\). If \(B^{-1/2} X B^{-1/2} \preceq t \cdot 1\), then \(t^{-1} \cdot X \preceq B\).
Conjugating the hypothesis by \(B^{1/2}\) preserves the order. On the left the two inverse square roots cancel, on the right \(t \cdot 1\) becomes \(t \cdot B\); dividing by \(t\) gives the claim.