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Given an idempotent p in a Banach algebra and following the study in [6] of p-invertibility, we consider here left p-invertibility, right p-invertibility and p-invertibility in the Calkin Algebra 𝒞(X), where X is a Banach space. Then we define and study left and right generalized Drazin invertibility and we characterize left and right Drazin invertible elements in the Calkin algebra. Globally, this leads to define and characterize the classes of P-Fredholm, pseudo B-Fredholm and weak B-Fredholm operators.

In 2004, Patrício and Puystjens characterized the relation between Drazin invertible elements (resp., Moore-Penrose invertible elements) of two semigroups pRp and pRp+1- p of a ring R for some idempotent (resp., projection) p ∈ R. In this paper, we consider the relevant result for pseudo core invertible elements of such two semigroups of a ring for some projection, which is then applied to characterize the relation between pseudo core invertible elements of the matrix semigroup AA†R m×mAA† +Im -AA† and the matrix semigroup A †ARn×nA †A+In -A †A, where A ∈ R m×n with A † existing. Also, similar equivalence involving DMP invertible elements is investigated.

Concave regularization methods provide natural procedures for sparse recovery. However, they are difficult to analyze in the highdimensional setting. Only recently a few sparse recovery results have been established for some specific local solutions obtained via specialized numerical procedures. Still, the fundamental relationship between these solutions such as whether they are identical or their relationship to the global minimizer of the underlying nonconvex formulation is unknown. The current paper fills this conceptual gap by presenting a general theoretical framework showing that, under appropriate conditions, the global solution of nonconvex regularization leads to desirable recovery performance; moreover, under suitable conditions, the global solution corresponds to the unique sparse local solution, which can be obtained via different numerical procedures. Under this unified framework, we present an overview of existing results and discuss their connections. The unified view of this work leads to a more satisfactory treatment of concave high-dimensional sparse estimation procedures, and serves as a guideline for developing further numerical procedures for concave regularization.

The invertibility of Wiener–Hopf plus Hankel operators W(a)+H(b) acting on the spaces L^p(R^+) , 1 p<ınfty is studied. If a and b belong to a subalgebra of L^ınfty(R) and satisfy the condition a(t) a(-t)=b(t) b(-t), tınR, we establish necessary and also sufficient conditions for the operators W(a)+H(b) to be one-sided invertible, invertible or generalized invertible. Besides, efficient representations for the corresponding inverses are given.

Let A and B be two group invertible matrices, we study the rank, the nonsingularity and the group invertibility of A − B, AA # − BB #], c₁A + c₂B, c₁A + c₂B + c₃AA#B where c₁, c₂ are nonzero complex numbers. Under some special conditions, the necessary and sufficient conditions of c₁A + c₂B + c₃AB and c₁A + c₂B + c₃AB + c₄BA to be nonsingular and group invertible are presented, which generalized some related results of Benítez, Liu, Koliha and Zuo [4, 17, 19, 25].

Let H be a complex infinite dimensional Hilbert space, B ( H ) be the algebra of all bounded linear operators acting on H , and H C ( H ) ¯ ( S C ( H ) ¯ ) be the norm closure of the class of all hypercyclic operators (supercyclic operators) in B ( H ). An operator T ∈ B ( H ) is said to be with hypercyclicity (supercyclicity) if T is in H C ( H ) ¯ ( S C ( H ) ¯ ) . Using a new spectrum defined from “consistent in invertibility”, this paper gives necessary and sufficient conditions that T is with a-Browder’s theorem or with a-Weyl’s theorem. Further, this paper gives a necessary and sufficient condition that T is a-isoloid, with a-Weyl’s theorem and with hypercyclicity (supercyclicity) concurrently. Also, the relations between that T is with hypercyclicity (supercyclicity) and that T is both with a-Weyl’s theorem and a-isoloid are discussed by means of the new spectrum.

The main objective of this paper is to review and promote the usefulness of generalized fractionally differenced Gegenbauer processes in time series and econometric research endeavours. In particular, theoretical and computational aspects centered around fractionally differenced Gegenbauer processes with long memory together with a number of interesting and elegant extensions will be discussed. In-depth conceptual developments and large scale simulation study results are presented for clarity and completeness. This survey highlights a number of gaps in the existing literature of this subject area and becomes a valuable reference source for time series practitioners.

In this paper, the multiplicative perturbation of group invertible operators is studied. We first establish necessary and sufficient conditions for the existence, along with explicit expressions, of the group inverse under multiplicative perturbation. Building on these results, we analyze the continuity of multiplicative perturbation and derive perturbation bounds for the group inverse. Furthermore, we extend our study to the group invertibility of linear combinations of operators.

We study the absolute penalized maximum partial likelihood estimator in sparse, high-dimensional Cox proportional hazards regression models where the number of time-dependent covariates can be larger than the sample size. We establish oracle inequalities based on natural extensions of the compatibility and cone invertibility factors of the Hessian matrix at the true regression coefficients. Similar results based on an extension of the restricted eigenvalue can be also proved by our method. However, the presented oracle inequalities are sharper since the compatibility and cone invertibility factors are always greater than the corresponding restricted eigenvalue. In the Cox regression model, the Hessian matrix is based on time-dependent covariates in censored risk sets, so that the compatibility and cone invertibility factors, and the restricted eigenvalue as well, are random variables even when they are evaluated for the Hessian at the true regression coefficients. Under mild conditions, we prove that these quantities are bounded from below by positive constants for time-dependent covariates, including cases where the number of covariates is of greater order than the sample size. Consequently, the compatibility and cone invertibility factors can be treated as positive constants in our oracle inequalities.