We characterize tauberian operators $T:L_1\left(\mu \right)\to Y$ in terms of the images of disjoint sequences and in terms of the image of the dyadic tree in $L_1[0,1]$. As applications, we show that the class of tauberian operators is stable under small norm perturbations and that its perturbation class coincides with the class of all weakly precompact operators. Moreover, we prove that the second conjugate of a tauberian operator $T:L_1\left(\mu \right)\to Y$ is also tauberian, and the induced operator $T̃:L_1\left(\mu \right)**/L_1\left(\mu \right)\to Y**/Y$ is an isomorphism into. Also, we show that $L_1\left(\mu \right)$ embeds...

A Banach space operator T ∈ has a left m-inverse (resp., an essential left m-inverse) for some integer m ≥ 1 if there exists an operator S ∈ (resp., an operator S ∈ and a compact operator K ∈ ) such that ${\sum }_{i=0}^m{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i}\right)S^{m-i}{T}^{m-i}=0$ (resp., ${\sum }_{i=0}^m{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i}\right)T^{m-i}{S}^{m-i}=K$). If $T_i$ is left $m_i$-invertible (resp., essentially left $m_i$-invertible), then the tensor product T₁ ⊗ T₂ is left (m₁ + m₂-1)-invertible (resp., essentially left (m₁ + m₂-1)-invertible). Furthermore, if T₁ is strictly left m-invertible (resp., strictly essentially left m-invertible), then...

We give an explicit description of a tensor norm equivalent on $C\left(K\right)\otimes F$ to the associated tensor norm ${\nu }_{qp}$ to the ideal of $(q,p)$-absolutely summing operators. As a consequence, we describe a tensor norm on the class of Banach spaces which is equivalent to the left projective tensor norm associated to ${\nu }_{qp}$.

We characterize Banach lattices on which every weak Banach-Saks operator is b-weakly compact.

The following properties of C[0,1] are proved here. Let T: C[0,1] → Y be a disjointness preserving bijection onto an arbitrary vector lattice Y. Then the inverse operator $T^{-1}$ is also disjointness preserving, the operator T is regular, and the vector lattice Y is order isomorphic to C[0,1]. In particular if Y is a normed lattice, then T is also automatically norm continuous. A major step needed for proving these properties is provided by Theorem 3.1 asserting that T satisfies some technical condition...

This paper will consider the closure of the set of operators which may be expressed as a sum of lattice homomorphisms whose range is contained in a Dedekind complete Banch lattice.

In this article we will illustrate how the Berezin transform (or symbol) can be used to study classes of operators on certain spaces of analytic functions, such as the Hardy space, the Bergman space and the Fock space. The article is organized according to the following outline. 1. Spaces of analytic functions 2. Definition and properties Berezin transform 3. Berezin transform and non-compact operators 4. Commutativity of Toeplitz operators 5. Berezin transform and Hankel or Toeplitz operators 6....

This paper studies the boundary behavior of the Berezin transform on the C*-algebra generated by the analytic Toeplitz operators on the Bergman space.

We establish $L^p$-estimates for the weighted Bergman projection on a nonsingular cone. We apply these results to the weighted Fock space with respect to the minimal norm in ℂⁿ.

We find necessary and sufficient conditions on radial weights w on the unit disc so that the Bergman type projections of Forelli-Rudin are bounded on L¹(w) and in the Herz spaces $K_p^q\left(w\right)$.