Indefinite Least Squares Problems and Oblique Projections

J. I. GIRIBET; A. MAESTRIPIERI; F. MARTÍNEZ PERÍA

Guanajuato, México

Workshop; International Workshop con Operator Theory and Applications; 2009

IWOTA - CIMAT

Given Hilbert spaces $mathcal{H}$ and $mathcal{K}$, a closed range operator $Cin L(mathcal{H},mathcal{K})$, a selfadjoint operator $Bin L(mathcal{K})$ and a vector $yin mathcal{K}$, we say that a vector $uinmathcal{H}$ is a $B$-least squares solution ($B$-LSS) of the equation $Cx=y$ if it satisfies: %egin{equation}label{BLSS} $ langle B(Cu-y),Cu-y angle =min_{xinmathcal{H}} langle B(Cx-y),Cx-y angle .$ %end{equation} In this work we study the existence of $B$-LSS and its relation with the set of (oblique) projections. Given a selfadjoint operator $B$ and a closed subspace $mathcal{S}$, the pair $(B,mathcal{S})$ is compatible if there exists a projection $Q$ with $R(Q)=mathcal{S}$ which is $B$-selfadjoint, i.e. selfadjoint with respect to the sesquilinear form $langle B.,. angle $. It is well known that the set of such projections $Q$ is an affine manifold, possibly empty. %the set $mathcal{P}(B,mathcal{S})={Q=Q^2: R(Q)=mathcal{S}, BQ=Q^*B}$ is not empty. It is well known that the set $mathcal{P}(B,mathcal{S})$ is an affine manifold, possible empty. We show that there exist $B$-LSS, for every $yinmathcal{K}$, if and only if the pair $(B,R(C))$ is compatible and $R(C)$ is $B$-nonegative, i.e. $langle Bx,x angle geq0$, for every $xin R(C)$. Furthermore we relate the set of $B$-LSS with the set $mathcal{P}(B,R(C))$. Also, a minimization problem among the $B$-LSS of the equation $Cx=y$ is presented. If $Ain L(mathcal{H})$ is an appropriate selfadjoint operator, we look for those $win mathcal{H}$ which are $B$-LSS of $Cx=y$ and satisfy: $langle w,w angle _{A} leq langle u,u angle _{A},$ for every $B$-LSS $uin mathcal{H}$ of $Cx=y$. A vector $winmathcal{H}$ satisfying the above conditions is called an $AB$-least squares solution ($AB$-LSS) of the equation $Cx=y$. We show that $win mathcal{H}$ is an $A B$-LSS of $Cx=y$ if and only if $w=(I-Q)C^dag Py$, where $P$ and $Q$ are appropriate $B$-selfadjoint and $A$-selfadjoint projections, respectively. In this case, the operator $D=(I-Q)C^dag Pin L(mathcal{K}, mathcal{H})$ can be seen as a ``weighted inverse´´ of $C$ because it is a solution of: $ CXC=C, XCX=X, A(XC)=(XC)^*A, B(CX)=(CX)^*B. $ Finally, we investigate a generalization of the $B$-LSS problem, making no assumptions on the of $R(C)$. In this case, there no longer exist $B$-LSS of the equation $Cx=y$. However, an analogous min-max problem can be stated.