On Numerical Radius Bounds Involving Generalized Aluthge Transform

In this paper, we establish some upper bounds of the numerical radius of a bounded linear operator S defined on a complex Hilbert space with polar decomposition S = U ∣ S ∣ , involving generalized Aluthge transform. These bounds generalize some bounds of the numerical radius existing in the literature. Moreover, we consider particular cases of generalized Aluthge transform and give some examples where some upper bounds of numerical radius are computed and analyzed for certain operators.

Comparative Study of Modified Mueller Matrix Transformation and Polar Decomposition Parameters for Transmission and Backscattering Tissue Polarimetries

Mueller matrix polarimetry is widely used in biomedical studies and applications, for it can provide abundant microstructural information about tissues. Recently, several methods have been proposed to decompose the Mueller matrix into groups of parameters related to specific optical properties which can be used to reveal the microstructural information of tissue samples more clearly and quantitatively. In this study, we thoroughly compare the differences among the parameters derived from the Mueller matrix polar decomposition (MMPD) and Mueller matrix transformation (MMT), which are two popular methods in tissue polarimetry studies and applications, while applying them on different tissue samples for both backscattering and transmission imaging. Based on the Mueller matrix data obtained using the setups, we carry out a comparative analysis of the parameters derived from both methods representing the same polarization properties, namely depolarization, linear retardance, fast axis orientation and diattenuation. IN particular, we propose several modified MMT parameters, whose abilities are also analyzed for revealing the information about the specific type of tissue samples. The results presented in this study evaluate the applicability of the original and modified MMT parameters, then give the suggestions for appropriate parameter selection in tissue polarimetry, which can be helpful for future biomedical and clinical applications.

Polar decompositions of quaternion matrices in indefinite inner product spaces

Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an $H$-polar decomposition are found. In the process, an equivalent to Witt's theorem on extending $H$-isometries to $H$-unitary matrices is given for quaternion matrices.

Unbounded Wiener–Hopf Operators and Isomorphic Singular Integral Operators

Some basics of a theory of unbounded Wiener–Hopf operators (WH) are developed. The alternative is shown that the domain of a WH is either zero or dense. The symbols for non-trivial WH are determined explicitly by an integrability property. WH are characterized by shift invariance. We study in detail WH with rational symbols showing that they are densely defined, closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are explicitly determined. Another topic concerns semibounded WH. There is a canonical representation of a semibounded WH using a product of a closable operator and its adjoint. The Friedrichs extension is obtained replacing the operator by its closure. The polar decomposition gives rise to a Hilbert space isomorphism relating a semibounded WH to a singular integral operator of Hilbert transformation type. This remarkable relationship, which allows to transfer results and methods reciprocally, is new also in the thoroughly studied case of bounded WH.

Some notes on complex symmetric operators

In this paper we show that every conjugation C on the Hardy-Hilbert space H 2 is of type C = T * 𝒥T, where T is an unitary operator and 𝒥 f ( z ) = f ( z ¯ ) ¯ \mathcal{J}f\left( z \right) = \overline {f\left( {\bar z} \right)} with f ∈ H 2. Moreover we prove some relations of complex symmetry between the operators T and |T|, where T = U |T| is the polar decomposition of bounded operator T ∈ ℒ(ℋ) on the separable Hilbert space ℋ.