arXiv subjects
Publications and source records attributed to Ilya M. Spitkovsky.
It is shown that the result of Tso-Wu on the elliptical shape of the numerical range of quadratic operators holds also for the essential numerical range. The latter is described quantitatively, and based on that sufficient conditions are established under which the c-numerical range also is an ellipse. Several examples are considered, including singular integral operators with the Cauchy kernel and composition operators.
A generalized word in two letters $A$ and $B$ is an expression of the form $W=A^{α_1}B^{β_1}A^{α_2}B^{β_2}... A^{α_N}B^{β_N}$ in which the exponents $α_i$, $β_i$ are nonzero real numbers. When independent positive definite matrices are substituted for $A$ and $B$, we are interested in whether $W$ necessarily has positive eigenvalues. This is known to be the case when N=1 and has been studied in case all exponents are positive by two of the authors. When the exponent signs are mixed, however, the situation is quite different (even for 2-by-2 matrices), and this is the focus of the present work.
Let G be a piecewise constant $n\times n$ matrix function which is defined on a smooth closed curve $Γ$ in the complex sphere and which has m jumps. We consider the problem of determining the partial indices of the factorization of the matrix function G in the space $L^p(Γ)$. We show that this problem can be reduced to a certain problem for systems of linear differential equations. Studying this related problem, we obtain some results for the partial indices for general n and m. A complete answer is given for n=2, m=4 and for n=m=3. One has to distinguish several cases. In some of these cases, the partial indices can be determined explicitly. In the remaining cases, one is led to two possibilities for the partial indices. The problem of deciding which is the correct possibility is equivalent to the description of the monodromy of n-th order linear Fuchsian differential equations with m singular points.