#### Location

Cupples I Room 115

#### Start Date

7-21-2016 4:00 PM

#### End Date

21-7-2016 4:20 PM

#### Description

Consider Hermitian matices $A, B, C$ such that $A + B = C$. Let $\{\lambda_j (A)\}$, $\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ be sequences of eigenvalues of $A, B$, and $C$ counting multiplicity, arranged in decreasing order. In 1962, A. Horn conjectured that the relations of $\{\lambda_j (A)\}$, $\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ can be characterized by a set of inequalities defined inductively. This conjecture was proved true by Klyachko and Knutson-Tao in the late 1990s. A related question, the multiplicative Horn problem, asks to describe the possible singular values of $AB$ when the singular values of $A$ and $B$ are given. This problem is fully solved in the case where $A$ and $B$ are invertible matrices as Klyachko showed that it is equivalent to the additive problem after taking logarithms. In this talk we will discuss the case when $A$ and $B$ are not necessarily invertible and its generalization to the von Neumann algebra setting. This is joint work with H. Bercovici, B. Collins, and K. Dykema.

Horn inequalities for singular values of products of operators

Cupples I Room 115

Consider Hermitian matices $A, B, C$ such that $A + B = C$. Let $\{\lambda_j (A)\}$, $\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ be sequences of eigenvalues of $A, B$, and $C$ counting multiplicity, arranged in decreasing order. In 1962, A. Horn conjectured that the relations of $\{\lambda_j (A)\}$, $\{\lambda_j (B)\}$, and $\{\lambda_j (C)\}$ can be characterized by a set of inequalities defined inductively. This conjecture was proved true by Klyachko and Knutson-Tao in the late 1990s. A related question, the multiplicative Horn problem, asks to describe the possible singular values of $AB$ when the singular values of $A$ and $B$ are given. This problem is fully solved in the case where $A$ and $B$ are invertible matrices as Klyachko showed that it is equivalent to the additive problem after taking logarithms. In this talk we will discuss the case when $A$ and $B$ are not necessarily invertible and its generalization to the von Neumann algebra setting. This is joint work with H. Bercovici, B. Collins, and K. Dykema.