Operator Theory, Complex Analysis and Applications Seminar 

Let $\mathscr{H}$ be a complex Hilbert space and let ${𝒮}_{\mathscr{H}}=\{x\in \mathscr{H};\parallel x\parallel =1\}$ be the unit sphere of $\mathscr{H}$. Every bounded linear operator $A$ on $\mathscr{H}$ defines a quadratic form as follows $$q_A:\begin{array}{c}{𝒮}_{\mathscr{H}}⟶ℂ\\ x\mapsto ⟨Ax,x⟩,\end{array}$$ where $⟨\cdot ,\cdot ⟩$ denotes the inner product of $\mathscr{H}$. The image of $q_A$ is the numerical range $W(A)\subset ℂ$ of $A$. It is not hard to see that the spectrum of an operator is a subset of the closure of the numerical range, which means that numerical ranges are a useful tool in locating the spectrum. Some classical results about numerical ranges will be presented; for instance, the Toeplitz-Hausdorff Theorem and the Hildebrandt's Theorem. The hyperreflexivity of sets of operators determined by the numerical range will be discussed, as well.

In the beginning of the XX century, Plemelj introduced the notion of factorization of matrix functions. If $A$ is a $n\times n$ regular matrix-function on the unit circle $𝕋$, by (left) factorization of $A$ relative to $𝕋$, we mean the following representation: $$A(t)=A_{+}(t)\Lambda (t)A_{-}(t),t\in 𝕋,$$ where $$\Lambda (t)=\mathrm{diag}(t^{{\kappa }_1},\dots ,t^{{\kappa }_n}),$$ with ${\kappa }_1\ge \dots \ge {\kappa }_n$, ${\kappa }_i\in \mathbb{Z}$ and $A_{+}^{\pm 1}$ ($A_{-}^{\pm 1}$) is analytic and regular in ${𝕋}_{+}$ (${𝕋}_{-}$). The matrix-function factorization find applications in many fields like diffraction theory, the theory of differential equations and the theory of singular integral operators.However, only for a few classes of matrices is known the explicit formulas for the factors of the factorization. In our talk we will show a new method to obtain a factorization of rational matrix-functions. The constructed method is based on the relation between the general solution of an homogeneous Riemann-Hilbert problem and a solution of a linear system of difference equations with constant coefficients. We also will provide some examples of factorization of rational matrix function, constructed with the developed factorization procedure.