arXiv subjects
Publications and source records attributed to Maximiliano Contino.
We study the class of those linear relations that can be factorized as products of idempotent relations. We provide several characterizations of this class, extending known factorization results for operators to the more general setting of linear relations.
We address the existence of non-trivial closed invariant subspaces of operators $T$ on Banach spaces whenever their square $T^2$ have or, more generally, whether there exists a polynomial $p$ with $\mbox{deg}(p)\geq 2$ such that the lattice of invariant subspaces of $p(T)$ is non-trivial. In the Hilbert space setting, the $T^2$-problem was posed by Halmos in the seventies and in 2007, Foias, Jung, Ko and Pearcy conjectured it could be equivalent to the \emph{Invariant Subspace Problem}.
Three approximation problems in Krein spaces are studied, namely the indefinite weighted least squares problem and the related problems of indefinite abstract splines and smoothing. In every case, we analyze if the problem has a solution for every point of the Krein space, the existence of a linear and continuous operator that maps each data point to its solution and when the associated operator problem considering the J-trace has a solution.
A multivalued projection is an idempotent linear relation with invariant domain. We characterize multivalued projections that are operator ranges (called semiclosed) and provide several formulae of them. Moreover, we study the decomposability and continuity of multivalued projections, and describe nilpotent relations.
Multivalued projections are applied to the study of weighted least squares solutions of linear relations equations (or inclusions) and some of its applications. To this end a matrix representation of multivalued projections with respect to the closure of their ranges is described.
A linear relation $E$ acting on a Hilbert space is idempotent if $E^2=E.$ A triplet of subspaces is needed to characterize a given idempotent: $(\mathrm{ran} \, E, \mathrm{ran}(I-E), \mathrm{dom}\, E),$ or equivalently, $(\mathrm{ker}(I-E), \mathrm{ker}\, E, \mathrm{mul} \, E).$ The relations satisfying the inclusions $E^2 \subseteq E$ (sub-idempotent) or $E \subseteq E^2$ (super-idempotent) play an important role. Lastly, the adjoint and the closure of an idempotent linear relation are studied.
We extend to infinite dimensional Hilbert spaces a celebrated result, due to B. Polyak, about the convexity of the joint image of quadratic functions. We give sufficient conditions which assure that the joint image is also closed. However, we show that, in general, the closedness part of Polyak's theorem does not hold in the infinite dimensional setting, even for quadratic functions generated by compact operators. We give some applications to S-lemma type results.
If a nonnegative selfadjoint linear relation $A$ in a Hilbert space and a closed subspace $\mathcal{S}$ are assumed to satisfy that the domain of $A$ is invariant under the orthogonal projector onto $\mathcal{S},$ then $A$ admits a particular matrix representation with respect to the decomposition $\mathcal{S} \oplus \mathcal{S}^{\perp}$. This matrix representation of $A$ is used to give explicit formulae for the Schur complement of $A$ on $\mathcal{S}$ as well as the $\mathcal{S}-$compression of $A$.
We characterize the semiclosed projections and apply them to compute the Schur complement of a selfadjoint operator with respect to a closed subspace. These projections occur naturally when dealing with weak complementability.
On finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class ${\mathcal L}^{+2}$ of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in ${\mathcal L}^{+2}$ are developed, and membership in ${\mathcal L}^{+2}$ among special classes, including algebraic and compact operators, is examined.
In this work we study weighted total least squares problems on infinite dimensional spaces. We show that in most cases this problem does not admit a solution (except in the trivial case) and then, we consider a regularization on the problem. We present necessary conditions for the regularized problem to have a solution. We also show that, by restricting the regularized minimization problem to special subsets, the existence of a solution may be assured.
We study three well-known minimization problems in Hilbert spaces: the weighted least squares problem and the related problems of abstract splines and smoothing. In each case we analyze the solvability of the problem for every point of the Hilbert space in the corresponding data set, the existence of an operator that maps each data point to its solution in a linear and continuous way and the solvability of the associated operator problem in a fixed p-Schatten norm.
Given a bounded selfadjoint operator W on a Krein space H and a closed subspace S of H, the Schur complement of W to S is defined under the hypothesis of weak complementability. A variational characterization of the Schur complement is given and the set of selfadjoint operators W admitting a Schur complement with these variational properties is shown to coincide with the set of S-weakly complementable selfadjoint operators.
Given B, C and W operators in the algebra L(H) of bounded linear operators on the Krein space H, the minimization problem min (BX - C)^#W(BX - C), for X in L(H), is studied when the weight W is selfadjoint. The analogous maximization and min-max problems are also considered. Complete answers to these problems and to those naturally associated to trace clase operators on Krein spaces are given.
Let $\mathcal{H}$ be a Hilbert space, $L(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$ and $W \in L(\mathcal{H})$ a positive operator. Given a closed subspace $\mathcal{S}$ of $\mathcal{H}$, we characterize the shorted operator $W_{/ \mathcal{S}}$ of $W$ to $\mathcal{S}$ as the maximum and as the infimum of certain sets, for the minus order $\stackrel{-}{\leq}.$ Also, given $A \in L(\mathcal{H})$ with closed range, we study the following operator approximation problem considering the minus order: $$ min_{\stackrel{-}{\leq}} \ \{(AX-I)^*W(AX-I) : X \in L(\mathcal{H}), \mbox{ subject to } N(A^*W)\subseteq N(X) \}. $$ We show that, under certain conditions, the shorted operator $W_{/R(A)}$ (of $W$ to the range of $A$) is the minimum of this problem and we characterize the set of solutions.
A Krein space H and bounded linear operators B, C on H are given. Then, some min and max problems about the operators (BX - C)^{#}(BX -C), where X runs over the space of all bounded linear operators on H, are discussed. In each case, a complete answer to the problem, including solvability conditions and characterization of the solutions, is presented. Also, an adequate decomposition of B is considered and the min-max problem is addressed. As a by-product the Moore-Penrose inverse of B is characterized as the only solution of a variational problem. Other generalized inverses are described in a similar fashion as well.
Let $\mathcal{H}$ be a Hilbert space, $L(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$ and $W \in L(\mathcal{H})$ a positive operator such that $W^{1/2}$ is in the p-Schatten class, for some $1 \leq p< \infty.$ Given $A, B \in L(\mathcal{H})$ with closed range and $C \in L(\mathcal{H}),$ we study the following weighted approximation problem: analize the existence of \begin{equation}\label{eqa1} \underset{X \in L(\mathcal{H})}{min}\Vert AXB-C \Vert_{p,W}, \ \ \ \ (1) \end{equation} where $\Vert X \Vert_{p,W}=\Vert W^{1/2}X \Vert_{p}.$ We also study the related operator approximation problem: analize the existence of \begin{equation} \label{eqa2} \underset{X \in L(\mathcal{H})}{min} (AXB-C)^{*}W(AXB-C), \ \ \ \ (2) \end{equation} where the order is the one induced in $L(\mathcal{H})$ by the cone of positive operators. In this paper we prove that the existence of the minimum of (2) is equivalent to the existence of a solution of the normal equation $A^*W(AXB-C)=0.$ We also give sufficient conditions for the existence of the minimum of (1) and we characterize the operators where the minimum is attained.
Let $\mathcal{H}$ be a Hilbert space, $L(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$ and $W \in L(\mathcal{H})$ a positive operator such that $W^{1/2}$ is in the p-Schatten class, for some $1 \leq p< \infty.$ Given $A \in L(\mathcal{H})$ with closed range and $B \in L(\mathcal{H}),$ we study the following weighted approximation problem: analize the existence of $$\underset{X \in L(\mathcal{H})}{min}\Vert AX-B \Vert_{p,W},$$ where $\Vert X \Vert_{p,W}=\Vert W^{1/2}X \Vert_{p}.$ In this paper we prove that the existence of this minimum is equivalent to a compatibility condition between $R(B)$ and $R(A)$ involving the weight $W,$ and we characterize the operators which minimize this problem as $W$-inverses of $A$ in $R(B).$