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Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

The kernel of a Toeplitz operator is nearly S∗-invariant where S∗ is the backward shift on the Hardy space of the unit disk. Recently, the question when the kernel of a finite rank perturbation of a Toeplitz operator is nearly S∗-invariant with finite defect is studied in Liang and Partington (Integral Equ Oper Theory 92, Paper No. 35, 2020) where the multiplicity of S is one and in Chattopadhyay et al. (Adv. Oper. Theory 6, Paper No. 49, 2021) where the multiplicity of S is finite. This question is answered affirmatively for several important classes of Toeplitz operators in Liang and Partington (2020), Chattopadhyay et al. (2021). In this paper we give a complete answer to this question even when S is of infinite multiplicity. Furthermore, our approach is general enough to cover related questions on almost S∗-invariant and almost S-invariant kernels and to include related operators such as Hankel operators and product of Toeplitz and Hankel operators.

This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space H 2 . The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly S ∗ -invariant subspace of H 2 , is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.

In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space which is a vectorial generalization of a result of Chalendar–Gallardo–Partington. Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space.

Let $H^{2}$ be the Hardy space over the bidisk. It is known that Hilbert–Schmidt invariant subspaces of $H^{2}$ have nice properties. An invariant subspace which is unitarily equivalent to some invariant subspace whose continuous spectrum does not coincide with $\\overline{\\mathbb{D}}$ is Hilbert–Schmidt. We shall introduce the concept of splittingness for invariant subspaces and prove that they are Hilbert–Schmidt.

In this paper, we prove that every bounded linear operator on a separable Hilbert space has a non-trivial invariant subspace. This answers the well-known invariant subspace problem.

Let $T = (T_1, \\ldots , T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\\mathcal{H}$. The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$. In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Let $n \\geq 2$, and let $\\mathcal{Q}_i$, $i = 1, \\ldots , n$, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\\mathbb {C}$. If $\\mathcal{Q}_i^{\\bot }$, $i = 1, \\ldots , n$, is a zero-based shift invariant subspace, then the multiplicity of the joint $M_{\\textbf {z}} = (M_{z_1}, \\ldots , M_{z_n})$-invariant subspace $(\\mathcal{Q}_1 \\otimes \\cdots \\otimes \\mathcal{Q}_n)^{\\perp }$ of the Dirichlet space or the Hardy space over the unit polydisc in $\\mathbb {C}^{n}$ is given by \\[ \\mbox{mult}_{M_{\\textbf{{z}}}|_{ (\\mathcal{Q}_1 \\otimes \\cdots \\otimes \\mathcal{Q}_n)^{{\\perp}}} (\\mathcal{Q}_1 \\otimes \\cdots \\otimes \\mathcal{Q}_n)^{{\\perp}} = \\sum_{i=1}^{n} (\\mbox{mult}_{M_z|_{\\mathcal{Q}_i^{{\\perp}}} (\\mathcal{Q}_i^{\\bot})) = n. \\]A similar result holds for the Bergman space over the unit polydisc.

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations.Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

We prove that the rank of a non-trivial co-doubly commuting submodule is 2. More precisely, let 𝜑, 𝜓 ∈ 𝐻∞(𝔻) be two inner functions. If 𝒬𝜑 = 𝐻²(𝔻)/𝜑𝐻²(𝔻) and 𝒬𝜓 = 𝐻²(𝔻)/𝜓𝐻² (𝔻), then rank ( Q φ ⊗ Q 𝜓 ) ⊥ = 2 An immediate consequence is the following: Let 𝒮 be a co-doubly commuting submodule of 𝐻²(𝔻²). Then rank 𝒮 = 1 if and only if 𝒮 = Φ𝐻²(𝔻²) for some one variable inner function Φ ∈ 𝐻∞(𝔻²). This answers a question posed by R. G. Douglas and R. Yang [Integral Equations Operator Theory 38(2000), pp207–221]

This paper is concerned with the study of invariant subspace problems for nonlinear operators on Banach spaces/algebras. Our study reveals that one faces unprecedented challenges such as lack of vector space structure and unbounded spectral sets when tackling invariant subspace problems for nonlinear operators via spectral information. To bypass some of these challenges, we modified an eigenvalue problem for nonlinear operators to cater for the structural properties of nonlinear operators and then established that nonlinear operators of finite type on a complex Banach algebra have nontrivial invariant subspaces.