Linear operators preserving the numerical range (radius) on triangular matrices

We characterize those linear operators on triangular or diagonal matrices preserving the numerical range or radius.     

Preservers of spectral radius, numerical radius, or spectral norm of the sum on nonnegative matrices

Let be the set of entrywise nonnegative n×n matrices. Denote by r(A) the spectral radius (Perron root) of . Characterization is obtained for maps such that r(f(A)+f(B))=r(A+B) for all . In particular, it is shown that such a map has the form

     

Schur-multiplicative maps preserving unitarily invariant norms

Characterizations are obtained for maps on real or complex matrices which preserve both the Schur (Hadamard) product and a given unitarily invariant norm.     

Spectral radius of Hadamard product versus conventional product for non-negative matrices

We prove an inequality for the spectral radius of products of non-negative matrices conjectured by X. Zhan. We show that for all n×n non-negative matrices A and B, ρ(A°B)?ρ((A°A)(B°B))^1/2?ρ(AB), in which ° represents the Hadamard product.     

Polynomial numerical hulls of some normal matrices

In this note, polynomial numerical hulls of matrices of the form _A1⊕i_A2$A_1\oplus i{A}_2$, where _A1$A_1$ and _A2$A_2$ are Hermitian, are characterized.     

Isometries of a generalized numerical radius

For 0<q<1, the q-numerical range is defined on the algebra M_n of all n×n complex matrices by

W_q(A)={x^∗Ay:x,y∈Cⁿ,∥x∥=∥y∥=1,〈y,x〉=q}.     

Multiplicative maps preserving the higher rank numerical ranges and radii

Let M_n be the semigroup of n×n complex matrices under the usual multiplication, and let S be different subgroups or semigroups in M_n including the (special) unitary group, (special) general linear group, the semigroups of matrices with bounded ranks. Suppose Λ_k(A) is the rank-k numerical range and r_k(A) is the rank-k numerical radius of A∈M_n. Multiplicative maps ?:S→M_n satisfying r_k(?(A))=r_k(A) are characterized. From these results, one can deduce the structure of multiplicative preservers of Λ_k(A).     

Schur multiplier norms for Loewner matrices

We study upper bounds on the Schur multiplier norm of Loewner matrices for concave and convex functions. These bounds then immediately lead to upper bounds on the ratio of Schatten q  -norms of commutators _{‖[A,f(B)]‖q}/_‖[A,B]‖q${\Vert [A,f(B)]\Vert }_q/{\Vert [A,B]\Vert }_q$. We also consider operator monotone functions, for which sharper bounds are obtained.     

k-Potence preserving maps without the linearity and surjectivity assumptions

Let M_n be the space of all n × n complex matrices, and let Γ_n be the subset of M_n consisting of all n × n k-potent matrices. We denote by Ψ_n the set of all maps on M_n satisfying A − λB ∈ Γ_n if and only if ?(A) − λ?(B) ∈ Γ_n for every A,B ∈ M_n and λ ∈ C. It was shown that ? ∈ Ψ_n if and only if there exist an invertible matrix P ∈ M_n and c ∈ C with c^k−1 = 1 such that either ?(A) = cPAP⁻¹ for every A ∈ M_n, or ?(A) = cPA^TP⁻¹ for every A ∈ M_n.     

Product numerical range in a space with tensor product structure

We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.     

Spaces of matrices without non-zero eigenvalues in their field of definition, and a question of Szechtman

Let V be a vector space of dimension n over any field F. Extreme values for the possible dimension of a linear subspace of End_F(V) with a particular property are considered in two specific cases. It is shown that if E₁ is a subspace of End_F(V) and there exists an endomorphism g of V, not in E₁, such that for every hyperplane H of V some element of E₁ agrees with g on H, then E₁ has dimension at least . This answers a question that was posed by Szechtman in 2003. It is also shown that a linear subspace of M_n(F) in which no element possesses a non-zero eigenvalue in F may have dimension at most . The connection between these two properties, which arises from duality considerations, is discussed.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Signature preserving linear maps of Hermitian matrices

The structure of a unital linear map on hermitian matrices with the property that it preserves the set of invertible hermitian matrices with fixed indefinite inertia is examined. It turns out that such a map is either a unitary similarity or a unitary similarity followed by a transposition (the case when the fixed inertia has equal number of positive and negative eigenvalues is excluded).     

Diagonals and numerical ranges of direct sums of matrices

For any n-by-n matrix A  , we consider the maximum number k=k(A)$k=k(A)$ for which there is a k-by-k compression of A   with all its diagonal entries in the boundary ∂W(A)$\partial W(A)$ of the numerical range W(A)$W(A)$ of A. If A   is a normal or a quadratic matrix, then the exact value of k(A)$k(A)$ can be computed. For a matrix A   of the form B⊕C$B\oplus C$, we show that k(A)=2$k(A)=2$ if and only if the numerical range of one summand, say, B is contained in the interior of the numerical range of the other summand C   and k(C)=2$k(C)=2$. For an irreducible matrix A  , we can determine exactly when the value of k(A)$k(A)$ equals the size of A  . These are then applied to determine k(A)$k(A)$ for a reducible matrix A   of size 4 in terms of the shape of W(A)$W(A)$.     

Some inequalities for the Hadamard product and the Fan product of matrices

If A and B are nonsingular M-matrices, a sharp lower bound on the smallest eigenvalue τ(A★B) for the Fan product of A and B is given, and a sharp lower bound on τ(A°B^-1) for the Hadamard product of A and B^-1 is derived. In addition, we also give a sharp upper bound on the spectral radius ρ(A°B) for nonnegative matrices A and B.     

An explicit Lipschitz constant for the joint spectral radius

In 2002, Wirth has proved that the joint spectral radius of irreducible compact sets of matrices is locally Lipschitz continuous as a function of the matrix set. In the paper, an explicit formula for the related Lipschitz constant is obtained.     

Complement preserving operators

An operatorTVV on a real inner product space is called complement preserving if, wheneverU is aT-invariant subspace ofV the orthogonal complementU is alsoT-invariant. In this note we obtain some results on such operators.