Complex Analysis and Operator Theory

We consider the resolvent algebra $${R_A=\{T\in\mathcal{L} (X):\sup_{m \geq 0}\|(1+\,mA)T(1+\,mA)^{-1}\| < \infty \}}$$ , and Deddens’ algebra $${B_A= \{T\in \mathcal{B}(H) : \sup_{n\geq 0}\|A^nTA^{-n}\|<\infty\}}$$ . It is shown that both R A and B A–I possess non-trivial invariant subspaces when A is an algebraic operator of degree 2. This assertion becomes stronger than the existence of a hyper-invariant subspace for R A whenever R A ≠ {A}′. Investigation of the relationship between these two algebras is addressed for different classes of operators A. Also, a complete characterization of the algebra R A when A is an algebraic operator is given. For the finite dimensional case, we present an elementary example showing that R A contains properly {A}′ whenever A has an eigenvalue other than zero.

The minimax principle for eigenvalues in gaps of the essential spectrum in the form presented by Griesemer et al. (Doc Math 4:275–283, 1999) is adapted to cover certain abstract perturbative settings with bounded or unbounded perturbations, in particular ones that are off-diagonal with respect to the spectral gap under consideration. This in part builds upon and extends the considerations in the author’s appendix to Nakić et al. (J Spectr Theory 10:843–885, 2020). Several monotonicity and continuity properties of eigenvalues in gaps of the essential spectrum are deduced, and the Stokes operator is revisited as an example.

In an infinite-dimensional Hilbert space we define and study special unbounded maximal dissipative operators and use them for constructions of abstract examples of families of densely defined closed symmetric operators whose squares have trivial domains.

In this paper, we discuss the necessary and sufficient conditions for a polynomial P(z) to have all its zeros inside the open unit disc. These results involve two associated polynomials namely, the derivative of the reciprocal polynomial of P(z) and the reciprocal of the derivative of P(z). We also derive some generalizations of the classical Theorem of Laguerre.

Let $$\Psi $$ , $$\Phi $$ be s.n. functions, $$p\ge 2,$$ and let $$\varphi $$ be an operator monotone function on $$[0,\infty )$$ such that $$\varphi (0)=0.$$ If are such that A and B are strictly accretive and then also and $$\begin{aligned}{} & {} \vert {\;\!\!\vert {AX\varphi {(B)}-\varphi {(A)}XB}\vert \;\!\!}\vert _\Psi \\{} & {} \qquad \le \left\| \,\!\sqrt{\varphi \Bigl ({\tfrac{A+A^*}{2}}\Bigr )-\tfrac{A+A^*}{2}\varphi ^\prime \Bigl ({\tfrac{A+A^*}{2}}\Bigr )} \Bigl ({\tfrac{A+A^*}{2}}\Bigr )^{-1}\!A(AX-XB)B\Bigl ({\tfrac{B+B^*}{2}}\Bigr )^{-1}\!\!\right. \\{} & {} \qquad \quad \left. \sqrt{\varphi \Bigl ({\tfrac{B+B^*}{2}}\Bigr )-\tfrac{B+B^*}{2}\varphi ^\prime \Bigl ({\tfrac{B+B^*}{2}}\Bigr )}\,\right\| _\Psi \,\!\!. \end{aligned}$$ under any of the following conditions: Alternative inequalities for $$\vert {\;\!\!\vert {\cdot }\vert \;\!\!}\vert _{\Phi ^{(p)}}$$ norms are also obtained.

We use Stokes’s theorem to establish an explicit and concrete connection between the Bergman and Szegő projections on the disc, the ball, and on strongly pseudoconvex domains.

This paper presents theoretical aspects of a unified generalization for the abstract theory of coherent state/voice transforms over homogeneous spaces of compact groups using operator theory. Let G be a compact group and H be a closed subgroup of G. Let G/H be the left coset space of H in G and $$\mu $$ be the normalized G-invariant measure on G/H associated to the Weil’s formula with respect to the probability measures of G, H. Let $$(\pi ,\mathcal {H}_\pi )$$ be a continuous unitary representation of G with non-zero mean over H. In this article, we introduce the generalized notion of coherent state/voice transform associated to $$\pi $$ on the Hilbert function $$L^2(G/H,\mu )$$ . We then study basic analytic properties of these transforms.

In this paper, we compute $$K$$ -groups $$\{K_{n}(C^{*}(x))\}_{n=0}^{\infty }$$ of the $$C^{*}$$ -subalgebra $$C^{*}(x)$$ of $$B(H),$$ generated by a single operator $$x,$$ where $$H$$ is a separable infinite dimensional Hilbert space, and $$B(H)$$ is the operator algebra consisting of all (bounded linear) operators on $$H.$$ These computations not only provide nice examples in $$K$$ -theory, but also characterize-and-classify projections in a $$C^{*}$$ -algebra generated by a single operator. The main result of this paper shows that: the $$K$$ -groups of $$C^{*}(x)$$ are completely characterized by those of $$C^{*}(q),$$ where $$q$$ is the positive-operator part of $$x$$ in the polar decomposition of $$x.$$

In this paper a new holomorphic extension theorem is presented using Clifford analysis.

Some properties and applications of meromorphic factorization of matrix functions are studied. It is shown that a meromorphic factorization of a matrix function G allows one to characterize the kernel of the Toeplitz operator with symbol G without actually having to previously obtain a Wiener–Hopf factorization. A method to turn a meromorphic factorization into a Wiener–Hopf one which avoids having to factorize a rational matrix that appears, in general, when each meromorphic factor is treated separately, is also presented. The results are applied to some classes of matrix functions for which the existence of a canonical factorization is studied and the factors of a Wiener–Hopf factorization are explicitly determined.