Related papers: Solution of Invariant Subspace Problem in the Hilb…
This paper deals with representing in concrete fashion those Hilbert spaces that are vector subspaces of the Hardy spaces $H^p(\bb D^n) \ (1\le p\le \infty)$ that remain invariant under the action of coordinate wise multiplication by an…
If $T$ is a polynomially bounded operator, $\mathcal M$ is an invariant subspace of $T$, $T|_{\mathcal M}$ is a unilateral shift and $T^*|_{\mathcal M^\perp}$ is subnormal, then $T$ has a nontrivial hyperinvariant subspace. If an operator…
Generalizing the algebra of motion-invariant differential operators on a symmetric space we study invariant operators on equivariant vector bundles. We show that the eigenequation is equivalent to the corresponding eigenequation with…
We present two alternative proofs of Mandrekar's theorem, which states that an invariant subspaces of the Hardy space on the bidisc is of Beurling type precisely when the shifts satisfy a doubly commuting condition. The first proof uses…
We consider the problem of learning a linear operator $\theta$ between two Hilbert spaces from empirical observations, which we interpret as least squares regression in infinite dimensions. We show that this goal can be reformulated as an…
We show that if A is a Hilbert-space operator, then the set of all projections onto hyperinvariant subspaces of A, which is contained in the von Neumann algebra vN(A) that is generated by A, is independent of the representation of vN(A),…
The classical Beurling-Helson-Lowdenslager theorem characterizes the shift-invariant subspaces of the Hardy space $H^{2}$ and of the Lebesgue space $L^{2}$. In this paper, which is self-contained, we define a very general class of norms…
We examine the properties of algebras of linear transformations that leave invariant all subspaces in a totally ordered lattice of subspaces of an arbitrary vector space. We compare our results with those that apply for the corresponding…
We give a short proof of the main result of our previous paper [2]: every Schmidt subspace of a Hankel operator is the image of a model space by an isometric multiplier. This class of subspaces is closely related to nearly $S^*$-invariant…
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…
We prove that Hilbert space is distortable and, in fact, arbitrarily distortable. This means that for all lambda >1 there exists an equivalent norm |.| on l_2 such that for all infinite dimensional subspaces Y of l_2 there exist x,y in Y…
In this paper, we use a new method to solve a long-standing problem. More specifically, we show that the Beurling-type theorem holds in the Bergman space $A^2_\alpha(D)$ for any $-1<\alpha < +\infty$. That is, every invariant subspace $H$…
Let $B(H)$ be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $H$. For $T \in B(H)$ and $\lambda \in \mathbb{C}$, let $H_{T}(\{\lambda\})$ denotes the local spectral subspace of $T$ associated…
We prove the existence of the invariant subspaces of some operators in a real Banach space. For example, linear isometries have invariant subspaces
The set of matrix tuples with invariant subspaces whose dimensions sum up to the dimension of the space, but which do not span the whole space form an algebraic hypersurface. We found the equation of this hypersurface. This generalizes…
The Invariant Subspace Problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota), the ISP may be solved by…
We investigate the structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group. We work with square integrable representations, and we show that they are those for which we can construct an…
Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise…
The invariant subspaces of the Hardy space on $H^2(\mathbb{D})$ of the unit disc are very well known however in several variables the structure of the invariant subspaces of the classical Hardy spaces is not yet fully understood. In this…
A convergent iterative process is constructed for solving any solvable linear equation in a Hilbert space.