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In this paper, we prove that for a right linear bounded operator on a quaternionic Hilbert space, the norm and the numerical radius are equal if and only if the norm and the spectral radius are equal. We also show that the spherical spectrum of a quaternionic bounded operator is included in the closure of its numerical range, and we show that the numerical range of an operator on a quaternionic Hilbert space is not necessarily convex. For a quaternionic bounded normal operator, we prove that the convex hull of the closure of its numerical range is equal to the convex hull of its spherical spectrum. Finally, we give some inequalities between the numerical radius, the spectral radius and the norm of a right linear bounded operator, and we prove also that the norm and the numerical radius of a quaternionic bounded hyponormal operator are equal. [ABSTRACT FROM AUTHOR]

In this note, we establish the cartesian decomposition of C-normal operator on a complex separable Hilbert space. Using this we give two characterizations of C-normal operators. These characterizations are useful in checking C-normality of an operator. We illustrate these results with examples. [ABSTRACT FROM AUTHOR]

The class of polynomially normal operators is a wider class than the class of all normal operators. Inspired by some interesting well known facts about normal operators and by some recent work, we present new properties of polynomially normal operators. Precisely, we prove that under certain conditions polynomially normal operators are Drazin or even group invertible and we also give necessary and sufficient conditions for a polynomially normal operator to have a closed range. In addition, we characterize polynomially normal operators with or without closed ranges applying the adequate operator matrix representations. Furthermore, we show that in some cases a polynomially normal operator can be written as a direct sum of a normal operator and a nilpotent operator. What is more, we state several examples to illustrate our results. [ABSTRACT FROM AUTHOR]

Let B (H) denote the algebra of all bounded linear operators acting on a complex Hilbert space H . For A , B ∈ B (H) , define the bimultiplication operator M 2 , A , B on the class of Hilbert–Schmidt operators by M 2 , A , B (X) = A X B. In this paper, we show that if B is normal, then c o (W 0 (A) W 0 (B)) ⊆ W 0 (M 2 , A , B) , where co stands for the convex hull and W 0 (.) denotes the maximal numerical range. If in addition, A is hyponormal, this inclusion becomes an equality. Some remarks about the maximal numerical range of the generalized derivation δ 2 , A , B on the class of Hilbert–Schmidt operators are also given. [ABSTRACT FROM AUTHOR]

Let (M , τ) be a semi-finite von Neumann algebra, L 0 (M) be the set of all τ-measurable operators, μ t (x) be the generalized singular number of x ∈ L 0 (M) and f : [ 0 , ∞) → [ 0 , ∞) be a concave function. We proved that if x 1 , x 2 , ... , x n are normal operators in L 0 (M) , then μ (f (| ∑ k = 1 n x k |)) is submajorized by μ (f (∑ k = 1 n | x k |)). As an application, we obtained that if x is a matrix of normal operators x i j in L 0 (M) , then μ (f (| x |)) is submajorized by μ (∑ i , j = 1 n f (| x i j |)). [ABSTRACT FROM AUTHOR]

Let σ n (H) denote the class of operators A acting on a complex Hilbert space H such that W 0 (A) = c o (λ ∈ σ (A) : λ = A ) , where W 0 (A) and σ (A) are the maximal numericl range, the spectrum of A, respectively, and co stands for the convex hull. In this paper, we show that the class σ n (H) contains hyponormal operators. We also show that if two operators A ∈ σ n (H) and B ∈ σ n (H ′) have the same closure of the numerical range or have the same spectrum, then they also have the same maximal numerical range. An affirmative answer to the question 2.16 in Fialkow [Elementary operators and applications. Proceedings of the International Workshop; Bleuberen; 1991. p. 55–113] is also given. [ABSTRACT FROM AUTHOR]

In this paper, we give new results for the p-numerical radii w p ⋅ of Hilbert space operators. It is shown, among other inequalities, that if A is a Hilbert space operator, which belongs to the Schatten p-class, then w p p (A) ≥ w p / 2 p / 2 (A 2) 2 p / 2 + A ∗ A + A A ∗ p / 2 p / 2 2 p + inf θ ∈ R Re ⁡ (e i θ A) p p − Im ⁡ (e i θ A) p p 2 and w p p (A) ≤ 2 p / 2 − 2 inf θ ∈ R Re ⁡ ((e i θ A) 2) p / 2 p / 2 + A ∗ A + A A ∗ p / 2 p / 2 4 for 4 ≤ p < ∞. Also, w p p (A) ≥ w p / 2 p / 2 A 2 4 + A ∗ A + A A ∗ p / 2 p / 2 2 p / 2 + 2 + inf θ ∈ R Re ⁡ (e i θ A) p p − Im ⁡ (e i θ A) p p 2 and w p p (A) ≤ inf θ ∈ R Re ⁡ ((e i θ A) 2) p / 2 p / 2 + A ∗ A + A A ∗ p / 2 p / 2 2 p / 2 for 2 ≤ p ≤ 4 , where ⋅ p is the Schatten p-norm. Applications of these inequalities to certain classes of operators are also given. [ABSTRACT FROM AUTHOR]

This paper deals with a conjecture posed by Abbas Salemi in 2011 (Banach J. Math. Anal.), claiming that for the spectrum of every bounded linear operator on a separable Hilbert space there exists an estimable total decomposition. We partially solve this problem, in the sense that, for the normal operator, there exists an estimable decomposition, under approximately unitary equivalence. [ABSTRACT FROM AUTHOR]

We introduce the class of quasi-2-isometric operators on Hilbert space. This class extends the classes of 2-isometric operators due to Agler and Stankus and quasi-isometries by S.M.Patel. An operator T on a complex Hilbert space is called quasi-2-isometry if In the present article, we give operator matrix representation of quasi-2-isometric operator in order to obtain spectral properties of this operator. [ABSTRACT FROM AUTHOR]

The matrix operator defined by the left-hand side of the Bevis–Hall–Hartwig equation(briefly, the BHH operator) is not linear in the space of complex matrices. However, it becomes a linear operator if every matrixis replaced by the pair of real matrices of the same size, namely, by its real and imaginary parts. Introducing an appropriate scalar product in the resulting real space, we find necessary and sufficient conditions for the BHH operator to be normal. [ABSTRACT FROM PUBLISHER]

In this study, we provide an algorithm for the boundary of the q-numerical range of a reducible matrix. An example is given to show that the q-numerical range cannot be derived from the classical numerical range of matrices. [ABSTRACT FROM AUTHOR]