Some Open Problems in the Theory of Subnormal Operators
نویسندگان
چکیده
Subnormal operators arise naturally in complex function theory, differential geometry, potential theory, and approximation theory, and their study has rich applications in many areas of applied sciences as well as in pure mathematics. We discuss here some research problems concerning the structure of such operators: subnormal operators with finite-rank self-commutator, connections with quadrature domains, invariant subspace structure, and some approximation problems related to the theory. We also present some possible approaches for the solution of these problems.
منابع مشابه
Polynomially hyponormal operators
A survey of the theory of k-hyponormal operators starts with the construction of a polynomially hyponormal operator which is not subnormal. This is achieved via a natural dictionary between positive functionals on specific convex cones of polynomials and linear bounded operators acting on a Hilbert space, with a distinguished cyclic vector. The class of unilateral weighted shifts provides an op...
متن کاملThe Lifting Problem for Hyponormal Pairs of Commuting Subnormal Operators
We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of RC, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality. Our tools include the use of 2-variable weighted...
متن کاملSome concavity properties for general integral operators
Let $C_0(alpha)$ denote the class of concave univalent functions defined in the open unit disk $mathbb{D}$. Each function $f in C_{0}(alpha)$ maps the unit disk $mathbb{D}$ onto the complement of an unbounded convex set. In this paper, we study the mapping properties of this class under integral operators.
متن کاملDouble-null operators and the investigation of Birkhoff's theorem on discrete lp spaces
Doubly stochastic matrices play a fundamental role in the theory of majorization. Birkhoff's theorem explains the relation between $ntimes n$ doubly stochastic matrices and permutations. In this paper, we first introduce double-null operators and we will find some important properties of them. Then with the help of double-null operators, we investigate Birkhoff's theorem for descreate $l^p$ sp...
متن کاملTheory of Hybrid Fractional Differential Equations with Complex Order
We develop the theory of hybrid fractional differential equations with the complex order $thetain mathbb{C}$, $theta=m+ialpha$, $0<mleq 1$, $alphain mathbb{R}$, in Caputo sense. Using Dhage's type fixed point theorem for the product of abstract nonlinear operators in Banach algebra; one of the operators is $mathfrak{D}$- Lipschitzian and the other one is completely continuous, we prove the exis...
متن کامل