We show for $2\le p<\infty$ and subspaces $X$ of quotients of $L_p$ with a $1$-unconditional finite-dimensional Schauder decomposition that $K(X,{\ell }_p)$ is an $M$-ideal in $L(X,{\ell }_p)$.

We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces $X$ and $Y$ the subspace of all compact operators $𝒦(X,Y)$ is an $M(r_1{r}_2,s_1{s}_2)$-ideal in the space of all continuous linear operators $ℒ(X,Y)$ whenever $𝒦(X,X)$ and $𝒦(Y,Y)$ are $M(r_1,s_1)$- and $M(r_2,s_2)$-ideals in $ℒ(X,X)$ and $ℒ(Y,Y)$, respectively, with $r_1+s_1/2>1$ and $r_2+s_2/2>1$. We also prove that the $M(r,s)$-ideal $𝒦(X,Y)$ in $ℒ(X,Y)$ is separably determined. Among others, our results complete and improve some well-known results...

The Mazur-Ulam theorem [15] has been formulated as two registrations: cluster bijective isometric -> midpoints-preserving Function of E, F; and cluster isometric midpoints-preserving -> Affine Function of E, F; A proof given by Jussi Väisälä [23] has been formalized.

We prove, among other things, that the space C[0,ω₂] has no countably norming Markushevich basis. This answers a question asked by G. Alexandrov and A. Plichko.

We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property (umap) in terms of “block unconditionality”. Then we focus on translation invariant subspaces $L_E^p\left(\right)$ and $C_E\left(\right)$ of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work shows that the spaces $p_E\left(\right)$, p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between $L_E^p\left(\right)$ and $L_E^{p+2}\left(\right)$. These...

We study the problem of whether ${}_w\left(ⁿE\right)$, the space of n-homogeneous polynomials which are weakly continuous on bounded sets, is an M-ideal in the space (ⁿE) of continuous n-homogeneous polynomials. We obtain conditions that ensure this fact and present some examples. We prove that if ${}_w\left(ⁿE\right)$ is an M-ideal in (ⁿE), then ${}_w\left(ⁿE\right)$ coincides with ${}_{w0}\left(ⁿE\right)$ (n-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property (M) and derive that if ${}_w\left(ⁿE\right){=}_{w0}\left(ⁿE\right)$ and (E) is an M-ideal in...

We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself, if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = $\overline{f^{\circ }\overline{\varphi }}$ for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z 1, ..., z n) = $({\lambda }_1{z}_{i_1},...,{\lambda }_n{z}_{i_n})$ for |λ j| = 1, 1 ≤ j ≤ n, and (i 1; ..., i n)is some permutation of the integers from...