ON THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS

In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of $\mathcal {L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm-closed convex hull, then we can guarantee the existence of an operator that does not attain its norm. This allows us to provide the following generalisation of results due to Holub and Mujica. If E is a reflexive space, F is an arbitrary Banach space and the pair $(E, F)$ has the (pointwise-)bounded compact approximation property, then the following are equivalent: (i) $\mathcal {K}(E, F) = \mathcal {L}(E, F)$ ; (ii) Every operator from E into F attains its norm; (iii) $(\mathcal {L}(E,F), \tau _c)^* = (\mathcal {L}(E, F), \left \Vert \cdot \right \Vert )^*$ , where $\tau _c$ denotes the topology of compact convergence. We conclude the article by presenting a characterisation of the Schur property in terms of norm-attaining operators.

The role of the algebraic structure in Wold-type decomposition

In recent works [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, Wold-type decompositions in Baer ∗ \ast -rings, Linear Algebra Appl. 539 2018, 117–133] and [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, The role of algebraic structure in the invariant subspace theory, Linear Algebra Appl. 583 2019, 102–118], the algebraic analogues of the three major decomposition theorems of Wold, Nagy–Foiaş–Langer and Halmos–Wallen were established in the larger category of Baer * {*} -rings. The results have their versions for commuting pairs in von Neumann algebras. In the corresponding proofs, both norm and weak operator topologies are heavily involved. In this work, ignoring topological structures, we give an algebraic approach to obtain them in Baer * {*} -rings.

Generalized Multipliers for Left-Invertible Operators and Applications

Generalized multipliers for a left-invertible operator T, whose formal Laurent series $$U_x(z)=\sum _{n=1}^\infty (P_ET^{n}x)\frac{1}{z^n}+\sum _{n=0}^\infty (P_E{T^{\prime *n}}x)z^n$$ U x ( z ) = ∑ n = 1 ∞ ( P E T n x ) 1 z n + ∑ n = 0 ∞ ( P E T ′ ∗ n x ) z n , $$x\in \mathcal {H}$$ x ∈ H actually represent analytic functions on an annulus or a disc are investigated. We show that they are coefficients of analytic functions and characterize the commutant of some left-invertible operators, which satisfies certain conditions in its terms. In addition, we prove that the set of multiplication operators associated with a weighted shift on a rootless directed tree lies in the closure of polynomials in z and $$\frac{1}{z}$$ 1 z of the weighted shift in the topologies of strong and weak operator convergence.

Some properties of D-weak operator topology

Let 𝓖D(𝓗) denote the generalized effect algebra consisting of all positive linear operators defined on a dense linear subspace D of a Hilbert space 𝓗. The D-weak operator topology (introduced by other authors) on 𝓖D(𝓗) is investigated. The corresponding closure of the set of bounded elements of 𝓖D(𝓗) is the whole 𝓖D(𝓗). The closure of the set of all unbounded elements of 𝓖D(𝓗) is also the set 𝓖D(𝓗). If Q is arbitrary unbounded element of 𝓖D(𝓗), it determines an interval in 𝓖D(𝓗), consisting of all operators between 0 and Q (with the usual ordering of operators). If we take the set of all bounded elements of this interval, the closure of this set (in the D-weak operator topology) is just the original interval. Similarly, the corresponding closure of the set of all unbounded elements of the interval will again be the considered interval.

Some Fixed Point Theory Results for the Interpolative Berinde Weak Operator

Partially inspired by [Erdal Karapinar, Ravi Agarwal and Hassen Aydi, Interpolative Reich-Rus-Ćirić type contractions on partial metric spaces, Mathematics 6 (2018), 256] and [V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum 9(1) (2004), 43-53], we introduce a concept of interpolative Berinde weak contraction, and obtain some existence theorems for mappings satisfying such a contractive definition, and some of its extensions.

Andô dilations and inequalities on non-commutative domains

We obtain intertwining dilation theorems for non-commutative regular domains 𝒟f and non-commutative varieties 𝒱J in B(𝓗)n, which generalize Sarason and Szőkefalvi-Nagy and Foiaş's commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain 𝒟f (respectively, variety 𝒱J ) as well as a Schur-type representation for the unit ball of the Hardy algebra associated with the variety 𝒱J. We provide Andô-type dilations and inequalities for bi-domains 𝒟f ×c 𝒟g consisting of all pairs (X,Y ) of tuples X := (X1,…, Xn1) ∊ 𝒟f and Y := (Y1,…, Yn2) ∊ 𝒟g that commute, i.e. each entry of X commutes with each entry of Y . The results are new, even when n1 = n2 = 1. In this particular case, we obtain extensions of Andô's results and Agler and McCarthy's inequality for commuting contractions to larger classes of commuting operators. All the results are extended to bi-varieties 𝒱J1×c 𝒱J2, where 𝒱J1 and 𝒱J2 are non-commutative varieties generated by weak-operator-topology-closed two-sided ideals in non-commutative Hardy algebras. The commutative case and the matrix case when n1 = n2 = 1 are also discussed.