Let P(ε) = P0 + A(ε) be a stochasticity preserving analytic perturbation of a stochastic matrix P0. We characterize the hybrid Cesaro limit lim 1 N(ε) Pk(ε), ε↓0 N(ε) ∑ where N(ε) ↑ ∞ as ε ↓ 0, when P0 has eigenvalues on the unit circle in the complex plane other than 1.

Spectral properties of integral operators on spaces of analytic functions on the unit disk of the complex plane have been studied since 1918. In this dissertation we determine spectral pictures and resolvent estimates for Ces`aro-like operators on the weighted Bergman spaces and show in particular that some of these operators are subdecomposable. Moreover, in a special case, we show that some of these operators are subnormal, some are normaloid, and some are subscalar. We also determine the spectrum and essential spectrum as well as resolvent estimates for a class of integral operators acting on Banach spaces of analytic functions on the unit disk, including the classical Hardy and weighted Bergman spaces, analytic Besov spaces as well as certain Dirichlet spaces and generalized Bloch spaces. Our results unify and extend recent work by Aleman and Persson, [4], Ballamoole, Miller and Miller, [6], and Albrecht and Miller, [3]. In [3], another class of integral operators were investigated in the setting of the analytic Besov spaces and the little Bloch space where the spectra, essential spectra together with one sided analytic resolvents in the Fredholm regions of these operators were obtained along with an explicit strongly decomposable operator extending one of these operator and simultaneously lifting the other. In this disseration, we extend this spectral analysis to nonseparable generalized Bloch spaces using a modification of a construction due to Aleman and Persson, [4].

This thesis is divided into three parts. In the first part we study mixed norm estimates for Riesz transforms associated with various differential operators. First we prove the mixed norm estimates for the Riesz transforms associated with Dunkl harmonic oscillator by means of vector valued inequalities for sequences of operators defined in terms of Laguerre function expansions. In certain cases, the result can be deduced from the corresponding result for Hermite Riesz transforms, for which we give a simple and an independent proof. The mixed norm estimates for Riesz transforms associated with other operators, namely the sub-Laplacian on Heisenberg group, special Hermite operator on C^d and Laplace-Beltrami operator on the group SU(2) are obtained using their L^pestimates and by making use of a lemma of Herz and Riviere along with an idea of Rubio de Francia. Applying these results to functions expanded in terms of spherical harmonics, we deduce certain vector valued inequalities for sequences of operators defined in terms of radial parts of the corresponding operators. In the second part, we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian ∆_κ on weighted L^P-spaces. In the general case, for the chaotic behavior of the Dunkl-heat semigroup on weighted L^p-spaces, we only have partial results, but in the case of the heat semigroup generated by the standard Laplacian, a complete picture of the chaotic behavior is obtained on the spaces L^p ( R^d,〖 (φ_iρ (x ))〗^2 dx) where φ_iρ the Euclidean spherical function is. The behavior is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar. In the last part, we study mixed norm estimates for the Cesáro means associated with Dunkl-Hermite expansions on〖 R〗^d. These expansions arise when one considers the Dunkl-Hermite operator (or Dunkl harmonic oscillator)〖 H〗_κ:=-Δ_κ+|x|^2. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Cesáro means for Laguerre expansions with shifted parameter. In order to obtain the latter, we develop an argument to extend these operators for complex values of the parameters involved and apply a version of Three Lines Lemma.