The Banach space $$\mathcal {P}({}^2X)$$ of 2-homogeneous polynomials on the X can be naturally embedded in $${\mathrm{Lip}_0}(B_X)$$ real-valued Lipschitz functions $$B_X$$ that vanish at 0. We investigate whether is a complemented subspace . This line research considered as polynomial counterpart to classical result by Joram Lindenstrauss, asserting {P}({}^1X)=X^*$$ for every X. Our main asse...

We prove that if X and Y are operator spaces such that every completely bounded operator from X into Y is completely compact and Z is a completely complemented subspace of X ⊕ Y , then there exists a completely bounded automorphism τ : X ⊕ Y → X ⊕ Y with completely bounded inverse such that τZ = X0 ⊕ Y0, where X0 and Y0 are completely complemented subspaces of X and Y , respectively. If X and Y...

A twisted sum Z of two Banach spaces X and Y is defined (see [11]) through a short exact sequence: 0 −→ X −→ Z −→ Y −→ 0. These short exact sequences in the category of (quasi-)Banach spaces are considered naturally in the investigation of three space properties (a property P in the category of quasi-Banach spaces is called a three space property if for every short exact sequence as above, Z ha...

A ternary ring of operators is an “off-diagonal corner” of a C∗-algebra and the predual of a ternary ring of operators (if it exists) is of the form pR∗q for some von Neumann algebra R and projections p and q in R. In this paper, we prove that a subspace of the predual of a ternary ring of operators is completely 1-complemented if and only if it is completely isometrically isomorphic to the pre...

The aim of the present article is to solve in the negative a well-known open problem raised by Bierstedt, Meise and Summers in [BMS1] (see also [BM1]). We construct a countable inductive limit of weighted Banach spaces of holomorphic functions, which is not a topological subspace of the corresponding weighted inductive limit of spaces of continuous functions. As a consequence the topology of th...

A sequence ${T_n}_{n=1}^{infty}$ of bounded linear  operators on a separable infinite dimensional Hilbert space $mathcal{H}$ is called subspace-diskcyclic with respect to the closed subspace $Msubseteq mathcal{H},$ if there exists a vector $xin mathcal{H}$ such that the disk-scaled orbit ${alpha T_n x: nin mathbb{N}, alpha inmathbb{C}, | alpha | leq 1}cap M$ is dense in $M$. The goal of t...

An element s of an (abstract) algebra A is a single element of A if asb = 0 and a, b ∈ A imply that as = 0 or sb = 0. Let X be a real or complex reflexive Banach space, and let B be a finite atomic Boolean subspace lattice on X, with the property that the vector sum K +L is closed, for every K,L ∈ B. For any subspace lattice D ⊆ B the single elements of Alg D are characterised in terms of a coo...