Given an ideal a in A[x1, . . . , xn] where A is a Noetherian integral domain, we propose an approach to compute the Krull dimension of A[x1, . . . , xn]/a, when the residue class ring is a free A-module. When A is a field, the Krull dimension of A[x1, . . . , xn]/a has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetheri...

The spectral decomposition of normal linear (bounded) operators and the polar decomposition of arbitrary linear (bounded) operators on Hilbert spaces have been interesting and technically useful results in operator theory [3, 9, 13, 20]. The development of the concept of von Neumann algebras on Hilbert spaces has shown that both these decompositions are always possible inside of the appropriate...

Let (X,⟨⋅,⋅⟩) be a Hilbert C∗-module over C∗-algebra A and let S(A) the set of states on A. In this paper, we first compute norm derivative for nonzero elements x y X as follows: limt→0+‖x+ty‖−‖x‖t=1‖x‖max{Reφ(⟨x,y⟩):φ∈S(A),φ(⟨x,x⟩)=‖x‖2}.We then apply it to characterize different concepts orthogonality in X. particular, present simpler proof classical characterization Birkhoff–James C∗-modules...

The theory of  continuous frames in Hilbert spaces is extended, by using the concepts of measure spaces, in order to get the results of a new application of operator theory.  The $K$-frames were  introduced by G$breve{mbox{a}}$vruta (2012) for Hilbert spaces to study atomic systems with respect to a bounded linear operator. Due to the structure of  $K$-frames, there are many differences between...

Let k be a field, and let R = k[x1, x2, x3]. Given a Hilbert function H for a cyclic module over R, we give an algorithm to produce a stable ideal I such that R/I has Hilbert function H and uniquely minimal graded Betti numbers among all R/J with the same Hilbert function, where J is another stable ideal in R. We also show that such an algorithm is impossible in more variables and disprove a re...

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the AlgL-module U is initiated, where L is a completely distributive subspace lattice on a Hilbert space H. Furthermore, as an application of single elements, we study module isomorphisms between norm closed AlgN -modules, where N is a nest, and obtain the following result: Sup...

It is well known that in the commutative case, i.e. for A = C(X) being a commutative C *-algebra, compact selfadjoint operators acting on the Hilbert C *-module H A (= continuous families of such operators K(x), x ∈ X) cannot be diagonalized inside this module but it becomes possible if we pass to a bigger module over a bigger W *-algebra L ∞ (X) = A ⊃ A which can be obtained from A by completi...

We associate a sheaf model to a class of Hilbert modules satisfying a natural finiteness condition. It is obtained as the dual to a linear system of Hermitian vector spaces (in the sense of Grothendieck). A refined notion of curvature is derived from this construction leading to a new unitary invariant for the Hilbert module. A division problem with bounds, originating in Douady’s privilege, is...