## Non-commutative inequalities

#### Event Title

Monotonicity in several non-commuting variables

Crow 206

#### Start Date

7-22-2016 2:30 PM

#### End Date

22-7-2016 2:50 PM

#### Description

Let $f : (a,b) → \mathbb{R}.$ The function f is said to be matrix monotone if $A \leq B$ implies $f(A) \leq f(B)$ for all pairs of like- sized self-adjoint matrices with spectrum in $(a,b)$. Classically, Charles Loewner showed that a bounded Borel function is matrix monotone if and only if it is analytic and extends to be a self-map of the upper half plane. The theory of matrix montonicity has profound consequences for any general theory of matrix inequalities. For example, it might seem surprising that $X \leq Y$ does not imply that $X^2 \leq Y^2$, which is a consequence of Loewner’s theorem. We will discuss commutative and noncommutative generalizations to several variables of Loewner’s theorem

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Jul 22nd, 2:30 PM Jul 22nd, 2:50 PM

Monotonicity in several non-commuting variables

Crow 206

Let $f : (a,b) → \mathbb{R}.$ The function f is said to be matrix monotone if $A \leq B$ implies $f(A) \leq f(B)$ for all pairs of like- sized self-adjoint matrices with spectrum in $(a,b)$. Classically, Charles Loewner showed that a bounded Borel function is matrix monotone if and only if it is analytic and extends to be a self-map of the upper half plane. The theory of matrix montonicity has profound consequences for any general theory of matrix inequalities. For example, it might seem surprising that $X \leq Y$ does not imply that $X^2 \leq Y^2$, which is a consequence of Loewner’s theorem. We will discuss commutative and noncommutative generalizations to several variables of Loewner’s theorem