Commutant lifting and factorization of reproducing kernels

Bi-Isometries and Commutant Lifting

In a previous paper, the authors obtained a model for a bi-isometry, that is, a pair of commuting isometries on complex Hilbert space. This representation is based on the canonical model of Sz. Nagy and the third author. One approach to describing the invariant subspaces for such a bi-isometry using this model is to consider isometric intertwining maps from another such model to the given one. ...

Commutant Lifting for Commuting Row Contractions

The commutant lifting theorem of Sz.Nagy and Foiaş [22, 21] is a central result in the dilation theory of a single contraction. It states that if T ∈ B(H) is a contraction with isometric dilation V acting on K ⊃ H, and TX = XT , then there is an operator Y with ‖Y ‖ = ‖X‖, V Y = Y V and PHY = XPH. This result is equivalent to Ando’s Theorem that two commuting contractions have a joint (power) d...

Reproducing kernels , Bergman kernels , Poisson kernels

Refinement of Reproducing Kernels

We continue our recent study on constructing a refinement kernel for a given kernel so that the reproducing kernel Hilbert space associated with the refinement kernel contains that with the original kernel as a subspace. To motivate this study, we first develop a refinement kernel method for learning, which gives an efficient algorithm for updating a learning predictor. Several characterization...

Duality by Reproducing Kernels

LetA be a determined or overdetermined elliptic differential operator on a smooth compact manifold X. Write A( ) for the space of solutions of the system Au= 0 in a domain X. Using reproducing kernels related to various Hilbert structures on subspaces of A( ), we show explicit identifications of the dual spaces. To prove the regularity of reproducing kernels up to the boundary of , we specify t...