Let A be a C*-algebra with unit 1 and let S be the state space of A, i.e., S= {jÎA*: j³0, j (1)=1}. For each a Î A, the C*-algebra Numerical range is defined byV (a): = {j (a): jÎS}.We prove that if V (a) is a disc with center at the origin, then ||a+a*||=||a-a*||.

Let $\mathcal{ B(H)}$ be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space $\mathcal H$. Let $A \in \mathcal{ B(H)}$ be a normaloid compact operator. In this note, we show that there exist $\lambda$ in $\sigma_{p}(A)$ the point spectrum of $A$ and a sequence $(x_{n})$ of unit vectors $x_{n}\in \h$ such that $$ \modu{\lambda}= \nor{A} \quad \text{and} \quad \displaystyle \lim_{n}\langle A^{k}x_{n}, x_{n} \rangle=\lambda^{k} \quad (k=1, 2, \dots). $$ As a consequence of the obtained result, we show that $$ \lambda^{k} \in \sigma_{p}(A^{k})\cap W_{max}(A^{k}) \quad (k=1, 2, \dots), $$ where $W_{max}(T)$ denotes the maximal Numerical range of an operator $T \in \mathcal{ B(H)}$.

Let H be a separable Hilbert space and let AÎB (H). In this note the notion of a total decomposition is introduced, and it is shown that sometimes the block Numerical ranges corresponding to a total decomposition approximate s (A), sometimes not.

Let V be an n-dimensional complex inner product space. Suppose H is a subgroup of the symmetric group of degree m, and x: H®C is an irreducible character (not necessarily linear). Denote by VX (H) the symmetry class of tensors associated with H and X. Let K (T)ÎEnd (VX (H)) be the operator induced by TÎEnd (V). The decomposable Numerical range WX (T) of T is a subset of the classical Numerical range W (K (T)) of K (T) defined as:WX (T) = {(K (T) x*, x*): x* is a decomposable unit tensor}.In this paper, we study the interplay between the geometric properties of WX (T) and the algebraic properties of T. In fact, we extend some of the results of [C. K. Li and A. Zaharia, Decomposable Numerical range on orthonormal decomposable tensors, Linear Algebra Appl. 308 (2000), no, 1-3, 139-152] and [C. K. Li and A. Zaharia, Induced operators on symmetry classes of tensors, Trans. Amer. Math. Soc. 354 (2002), no. 2, 807-836], to non-linear irreducible characters.

We introduce the $\alpha-$higher rank form of the matrix Numerical range, which is a special case of the matrix polynomial version of higher rank Numerical range. We also, investigate some algebraic and geometrical properties of this set for general and nilpotent matrices. Some examples to confirm the results are brought.

The main results of this paper are generalizations of classical results from the Numerical range to the block Numerical range. A di erent and simpler proof for the Perron-Frobenius theory on the block Numerical range of an irreducible nonnegative matrix is given. In addition, the Wielandt’s lemma and the Ky Fan’s theorem on the block Numerical range are extended.