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We obtain two matrix inequalities on the off-diagonal block of 2×2 block positive semidefinite matrices and the geometric mean of its diagonal blocks. They improve some results of Lee [Linear Algeb...

Inequalities on partial traces of positive semidefinite matrices are studied. Extensions of several existing inequalities on the determinant of partial traces are then obtained. Particularly, we improve a determinantal inequality given by Lin [Canad. Math. Bull. 59(2016)].

We study unitarily invariant norm inequalities for sector block matrices and obtain new inequalities which extend some recent results of Jabbarzadeh and Kaleibary (2020) and generalize a norm inequality due to Lin and Zhou (2013). At the same time, we refine an inequality raised by Zhang (2015).

Let $$\mathbf{A}= (A_1, \ldots , A_m)$$ , where $$A_1, \ldots , A_m$$ are $$n\times n$$ real matrices. The real joint (p, q)-matricial range of $$\mathbf{A}$$ , $${\varLambda }^{{\mathbb {R}}}_{p,q}(\mathbf{A})$$ , is the set of m-tuple of $$q\times q$$ real matrices $$(B_1, \ldots , B_m)$$ such that $$(X^*A_1X, \ldots , X^*A_mX) = (I_p\otimes B_1, \ldots , I_p \otimes B_m)$$ for some real $$n \times pq$$ matrix X satisfying $$X^*X = I_{pq}$$ . It is shown that if n is sufficiently large, then the set $${\varLambda }^{{\mathbb {R}}}_{p,q}(\mathbf{A})$$ is non-empty and star-shaped. The result is extended to bounded linear operators acting on a real Hilbert space $${{{\mathcal {H}}}}$$ , and used to show that the joint essential (p, q)-matricial range of $$\mathbf{A}$$ is always compact, convex, and non-empty. Similar results for the joint congruence matricial ranges on complex operators are also obtained.

Let Φ : Mn → Mn be a unital trace preserving completely positive map and A ∈ Mn be a positive definite matrix. Weak log-majorization and weak majorization between Φ(A) and A are studied. Determinantal inequalities between Φ(A) and A are obtained as a consequence. By considering special classes of unital trace preserving completely positive map, some known matrix inequalities such as Fischer’s inequality are rediscovered. An affirmative answer to a question of Tam and Zhang in 2019 is given.

Let ${\bf A} = (A_1, \dots, A_m)$ be an $m$-tuple of bounded linear operators acting on a Hilbert space ${\cal H}$. Their joint $(p,q)$-matricial range $\Lambda_{p,q}({\bf A})$ is the collection of $(B_1, \dots, B_m) \in {\bf M}_q^m$, where $I_p\otimes B_j$ is a compression of $A_j$ on a $pq$-dimensional subspace. This definition covers various kinds of generalized numerical ranges for different values of $p,q,m$. In this paper, it is shown that $\Lambda_{p,q}({\bf A})$ is star-shaped if the dimension of $\cal H$ is sufficiently large. If $\dim {\cal H}$ is infinite, we extend the definition of $\Lambda_{p,q}({\bf A})$ to $\Lambda_{\infty,q}({\bf A})$ consisting of $(B_1, \dots, B_m) \in {\bf M}_q^m$ such that $I_\infty \otimes B_j$ is a compression of $A_j$ on a closed subspace of ${\cal H}$, and consider the joint essential $(p,q)$-matricial range $$\Lambda^{ess}_{p,q}({\bf A}) = \bigcap \{ {\bf cl}(\Lambda_{p,q}(A_1+F_1, \dots, A_m+F_m)): F_1, \dots, F_m \hbox{ are compact operators}\}.$$ Both sets are shown to be convex, and the latter one is always non-empty and compact., Comment: 14 pages

Thompson obtained necessary and sufficient conditions on the existence of a real square matrix with prescribed diagonal elements, prescribed singular values and non-negative determinant. Th...

Let H n be the set of all n × n Hermitian matrices and H n m be the set of all m -tuples of n × n Hermitian matrices. For A = ( A 1 , . . . , A m ) ∈ H n m and for any linear map L : H n m → R l , we define the L -numerical range and the L -numerical radius of A by W L ( A ) : = { L ( U ⁎ A 1 U , . . . , U ⁎ A m U ) : U ∈ C n × n , U ⁎ U = I n } and r L ( A ) : = max ⁡ { x x t : x ∈ W L ( A ) } , respectively. They can be regarded as a generalization of the C -numerical range and the C -numerical radius, respectively, of n × n complex matrices. In this paper, interplay between the geometric properties of W L ( A ) and algebraic properties of linear maps L : H n m → R l and A ∈ H n m will be studied. We shall give a characterization of L such that r L ( ⋅ ) defines a norm on H n m . Furthermore, we shall also characterize linear maps L : H n m → R l and A ∈ H n m in which W L ( A ) is a singleton set. More generally, the characterization of linear maps L : H n m → R l and A ∈ H n m in which W L ( A ) lies inside a k -dimensional affine subspace of R l will be given.

Let H n be the set of all n × n Hermitian matrices and H n m be the set of all m-tuples of n × n Hermitian matrices. For A = ( A 1 , . . . , A m ) ∈ H n m and for any linear map L : H n m → R l , we define the L-numerical range of A by W L ( A ) : = { L ( U ⁎ A 1 U , . . . , U ⁎ A m U ) : U ∈ C n × n , U ⁎ U = I n } . In this paper, we prove that if l ≤ 3 , n ≥ l and A 1 , . . . , A m are simultaneously unitarily diagonalizable, then W L ( A ) is star-shaped with star center at L ( tr A 1 n I n , . . . , tr A m n I n ) .

Let A ∈ R n × n and SO n : = { U ∈ R n × n : U U t = I n , det U > 0 } be the set of n × n special orthogonal matrices. Define the (real) special orthogonal orbit of A by O ( A ) : = { U A V : U , V ∈ SO n } . In this paper, we show that the linear image of O ( A ) is star-shaped with respect to the origin for arbitrary linear maps L : R n × n → R l if n ≥ 2 l − 1 . In particular, for linear maps L : R n × n → R 2 and when A has distinct singular values, we study B ∈ O ( A ) such that L ( B ) is a boundary point of L ( O ( A ) ) . This gives an alternative proof of a result by Li and Tam on the convexity of L ( O ( A ) ) for linear maps L : R n × n → R 2 .