Catalin Badea

Catalin Badea, Bernhard Beckermann

This is a survey about spectral sets, to appear in the second edition of Handbook of Linear Algebra (L. Hogben, ed.). Spectral sets and K-spectral sets, introduced by John von Neumann, offer a possibility to estimate the norm of functions of matrices in terms of the sup-norm of the function. Examples of such spectral sets include the numerical range or the pseudospectrum of a matrix, discussed in Chapters 16 and 18. Estimating the norm of functions of matrices is an essential task in numerous fields of pure and applied mathematics, such as (numerical) linear algebra, functional analysis, and numerical analysis. More specific examples include probability, semi-groups and existence results for operator-valued differential equations, the study of numerical schemes for the time discretization of evolution equations, or the convergence rate of GMRES (Section 41.7). The notion of spectral sets involves many deep connections between linear algebra, operator theory, approximation theory, and complex analysis.

Catalin Badea, Sophie Grivaux, Vladimir Muller

A generalization of the cosine of the Friedrichs angle between two subspaces to a parameter associated to several closed subspaces of a Hilbert space is given. This parameter is used to analyze the rate of convergence in the von Neumann-Halperin method of cyclic alternating projections. General dichotomy theorems are proved, in the Hilbert or Banach space situation, providing conditions under which the alternative QUC/ASC (quick uniform convergence versus arbitrarily slow convergence) holds. Several meanings for ASC are proposed.

Catalin Badea, David Seifert

Given $N\ge2$ closed subspaces $M_1,\dotsc, M_N$ of a Hilbert space $X$, let $P_k$ denote the orthogonal projection onto $M_k$, $1\le k\le N$. It is known that the sequence $(x_n)$, defined recursively by $x_0=x$ and $x_{n+1}=P_N\cdots P_1x_n$ for $n\ge0$, converges in norm to $P_Mx$ as $n\to\infty$ for all $x\in X$, where $P_M$ denotes the orthogonal projection onto $M=M_1\cap\dotsc\cap M_N$. Moreover, the rate of convergence is either exponentially fast for all $x\in X$ or as slow as one likes for appropriately chosen initial vectors $x\in X$. We give a new estimate in terms of natural geometric quantities on the rate of convergence in the case when it is known to be exponentially fast. More importantly, we then show that even when the rate of convergence is arbitrarily slow there exists, for each real number $α>0$, a dense subset $X_α$ of $X$ such that $\|x_n-P_Mx\|=o(n^{-α})$ as $n\to\infty$ for all $x\in X_α$. Furthermore, there exists another dense subset $X_\infty$ of $X$ such that, if $x\in X_\infty$, then $\|x_n-P_Mx\|=o(n^{-α})$ as $n\to\infty$ for all $α>0$. These latter results are obtained as consequences of general properties of Ritt operators. As a by-product, we also strengthen the unquantified convergence result by showing that $P_M x$ is in fact the limit of a series which converges unconditionally.

Catalin Badea, David Seifert

We present an extension of our earlier work [Ritt operators and convergence in the method of alternating projections, J. Approx. Theory, 205:133-148, 2016] by proving a general asymptotic result for orbits of an operator acting on a reflexive Banach space. This result is obtained under a condition involving the growth of the resolvent, and we also discuss conditions involving the location and the geometry of the numerical range of the operator. We then apply the general results to some classes of iterative projection methods in approximation theory, such as the Douglas-Rachford splitting method and, under suitable geometric conditions either on the ambient Banach space or on the projection operators, the method of alternating projections.