We introduce the B-spline interpolation problem corresponding to a C∗-valued sesquilinear form on Hilbert C∗-module and study its basic properties as well uniqueness of solution. first in case when is self-dual. Passing setting W∗-modules, we present our main result by characterizing spline for extended has Finally, solutions C∗-modules over C∗-ideals W∗-algebras are extensively discussed.

It is well-known that if T is a Dm–Dn bimodule map on the m×n complex matrices, then T is a Schur multiplier and ‖T‖cb = ‖T‖. If n = 2 and T is merely assumed to be a right D2-module map, then we show that ‖T‖cb = ‖T‖. However, this property fails if m ≥ 2 and n ≥ 3. For m ≥ 2 and n = 3, 4 or n ≥ m2 we give examples of maps T attaining the supremum C(m,n) = sup{‖T‖cb : T a right Dn-module map o...

Let κ be an U-invariant reproducing kernel and let H (κ) denote the reproducing kernel Hilbert C[z1, . . . , zd]-module associated with the kernel κ. Let Mz denote the d-tuple of multiplication operators Mz1 , . . . ,Mzd on H (κ). For a positive integer ν and d-tuple T = (T1, . . . , Td), consider the defect operator

Let κ be an U-invariant reproducing kernel and let H (κ) denote the reproducing kernel Hilbert C[z1, . . . , zd]-module associated with the kernel κ. Let Mz denote the d-tuple of multiplication operators Mz1 , . . . ,Mzd on H (κ). For a positive integer ν and d-tuple T = (T1, . . . , Td), consider the defect operator

Let H be a Hilbert C-module over a matrix algebra A. It is proved that any function T : H → H which preserves the absolute value of the (generalized) inner product is of the form Tf = φ(f)Uf (f ∈ H), where φ is a phase-function and U is an A-linear isometry. The result gives a natural extension of Wigner’s classical unitary-antiunitary theorem for Hilbert modules. 1991 Physics and Astronomy Cla...

It is well known that a function K : Ω × Ω → L(Y) (where L(Y) is the set of all bounded linear operators on a Hilbert space Y) being (1) a positive kernel in the sense of Aronszajn (i.e. ∑N i,j=1〈K(ωi, ωj)yj , yi〉 ≥ 0 for all ω1, . . . , ωN ∈ Ω, y1, . . . , yN ∈ Y, and N = 1, 2, . . . ) is equivalent to (2) K being the reproducing kernel for a reproducing kernel Hilbert space H(K), and (3) K ha...