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Let A and B be $ n\times n $ n × n positive semidefinite matrices, $ f:[0,\infty)\rightarrow \lbrack 0,\infty) $ f : [ 0 , ∞) → [ 0 , ∞) be an increasing convex function with $ f\left (0\right) =0 $ f (0) = 0 , and let $ \Phi : \mathbb {M}_{n}(\mathbb {C})\rightarrow \mathbb {M}_{n}(\mathbb {C}) $ Φ : M n (C) → M n (C) be a positive unital linear map. It is shown that $$\begin{align*} & \prod_{j=1}^{k} s_{j}\left(f^{4}\left(\Phi \left(\frac{A+B}{2}\right) \right) \right)\\ & \quad \leq 2^{-4} \prod_{j=1}^{k} \left(s_{j}\left(\Phi^{2}\left(f^{2}\left(A\right) \right) +\Phi \left(f^{2}\left(A\right) \right) \right) +\left\Vert \Phi \left(f^{2}\left(B\right) \right) \right\Vert +1\right) \\ & \quad\quad \times \left(s_{j}\left(\Phi^{2}\left(f^{2}\left(B\right) \right) +\Phi \left(f^{2}\left(B\right) \right) \right) +\left\Vert \Phi \left(f^{2}\left(A\right) \right) \right\Vert +1\right) \end{align*}$$ ∏ j = 1 k s j (f 4 (Φ (A + B 2))) ≤ 2 − 4 ∏ j = 1 k (s j (Φ 2 (f 2 (A)) + Φ (f 2 (A))) + ‖ Φ (f 2 (B)) ‖ + 1) × (s j (Φ 2 (f 2 (B)) + Φ (f 2 (B))) + ‖ Φ (f 2 (A)) ‖ + 1) for $ k=1,2,\ldots,n $ k = 1 , 2 , ... , n , where $ s_{j}\left (T\right) $ s j (T) and $ \left \Vert T\right \Vert $ ‖ T ‖ are the $ j{\rm th} $ j th singular value and the spectral norm of T, respectively. Applications of our results to sectorial matrices are given. [ABSTRACT FROM AUTHOR]

Let Ψ : ⨂ i = 1 d H n i → ⨂ i = 1 d H n i be a linear map on the Kronecker product of spaces of Hermitian matrices H n i of size n i ≥ 3. (If d=1, we identify ⨂ i = 1 d H n i with H n 1 .) We establish a condition under which Ψ adj ⨂ i = 1 d A i = adj Ψ ⨂ i = 1 d A i if and only if det Ψ ⨂ i = 1 d A i = det ⨂ i = 1 d A i for all ⨂ i = 1 d A i ∈ ⨂ i = 1 d H n i . Then for d ∈ { 1 , 2 } , we apply this fact to characterize maps Ψ : ⨂ i = 1 d H n i → ⨂ i = 1 d H n i such that Ψ adj ⨂ i = 1 d A i = adj Ψ ⨂ i = 1 d A i ⨂ i = 1 d with some mild conditions. [ABSTRACT FROM AUTHOR]

For a positive integer, letbe the set ofcomplex matrices. Supposeare positive integers and. Denote bythe-numerical range of a matrix. It is shown that a linear mapsatisfiesfor allandif and only if there is a unitarysuch that one of the following holds.   For all , ,     and for all , ,where (1)is the identity mapor the transposition map, or (2)andhas the formor. [ABSTRACT FROM PUBLISHER]

We study the maps on complex Hermitian and complex symmetric matrices which preserve zeros of a Jordan product. [ABSTRACT FROM AUTHOR]