Let U be an open subset of a separable Banach space. Let ℱ be the collection of all holomorphic mappings f from the open unit disc 𝔻 ⊂ ℂ into U such that f(𝔻) is dense in U. We prove the lineability and density of ℱ in appropriate spaces for different choices of U.
In the paper existence and uniqueness results for the linear differential system on the interval [0,1] with distributional coefficients and solutions from the space of regulated functions are obtained.
The paper deals with the linear differential equation (0.1) with distributional coefficients and solutions from the space of regulated functions. Our aim is to get the basic existence and uniqueness results for the equation (0.1) and to generalize the known results due to F. V. Atkinson [At], J. Ligeza [Li1]-[Li3], R. Pfaff ([Pf1], [Pf2]), A. B. Mingarelli [Mi] as well as the results from the paper [Pe-Tv] concerning the equation (0.1).
In set theory without the Axiom of Choice ZF, we prove that for every commutative field , the following statement : “On every non null -vector space, there exists a non null linear form” implies the existence of a “-linear extender” on every vector subspace of a -vector space. This solves a question raised in Morillon M., Linear forms and axioms of choice, Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421-431. In the second part of the paper, we generalize our results in the case of spherically...
Let be a complex Hilbert space, a positive operator with closed range in and the sub-algebra of of all -self-adjoint operators. Assume onto itself is a linear continuous map. This paper shows that if preserves -unitary operators such that then defined by is a homomorphism or an anti-homomorphism and for all , where and is the Moore-Penrose inverse of . A similar result is also true if preserves -quasi-unitary operators in both directions such that there exists an...
Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. For an operator T in B(H), let σg(T) denote the generalized spectrum of T. In this paper, we prove that if φ: B(H) → B(H) is a surjective linear map, then φ preserves the generalized spectrum (i.e. σg(φ(T)) = σg(T) for every T ∈ B(H)) if and only if there is A ∈ B(H) invertible such that either φ(T) = ATA-1 for every T ∈ B(H), or φ(T) = ATtrA-1 for every T ∈ B(H). Also, we...