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In this article we prove a conjecture formulated by A. B. Sørensen in 1991 on the maximal number of F q 2 \mathbb {F}_{q^2} -rational points on the intersection of a non-degenerate Hermitian surface and a surface of degree d ≤ q . d \le q.

Our goal is to combine the techniques of Xiaokui Yang, Valentino Tosatti, and others to establish a Liouville-type result for almost complex manifolds. The transition to the non-integrable setting is delicate, so we will devote a section to discuss the key differences, and another to introduce the tools we will be using. Afterwards, we present a proof of our main theorem.

We explore the regularity of the roots of Garding hyperbolic polynomials and real stable polynomials. As an application we obtain new regularity results of Sobolev type for the eigenvalues of Hermitian matrices and for the singular values of arbitrary matrices. These results are optimal among all Sobolev spaces., 13 pages; some details added and typos corrected, accepted for publication in Proc. Amer. Math. Soc

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a representation of $sl(2,\mathbb{C})$, generalizing the well known structure on the harmonic forms of compact K\"ahler manifolds. Some topological implications are deduced., Comment: 7 pages, to appear in Proc. AMS

We give several versions of Siu’s ∂ ∂ ¯ \partial \overline {\partial } -formula for maps from a strictly pseudoconvex pseudo-Hermitian manifold ( M 2 m + 1 , θ ) (M^{2m+1}, \theta ) into a Kähler manifold ( N n , g ) (N^n, g) . We also define and study the notion of pseudo-Hermitian harmonicity for maps from M M into N N . In particular, we prove a CR version of the Siu Rigidity Theorem for pseudo-Hermitian harmonic maps from a pseudo-Hermitian manifold with vanishing Webster torsion into a Kähler manifold having strongly negative curvature.

Let p p be an odd prime. Let F F be the function field of a p p -adic curve. Let A A be a central simple algebra of period 2 over F F with an involution σ \sigma . There are known upper bounds for the u u -invariant of hermitian forms over ( A , σ ) (A, \sigma ) . In this article we compute the exact values of the u u -invariant of hermitian forms over ( A , σ ) (A, \sigma ) .

We construct new covers of the Suzuki and Ree curves which are maximal with respect to the Hasse-Weil bound over suitable finite fields. These covers are analogues of the Giulietti-Korchm\'aros curve, which covers the Hermitian curve and is maximal over a base field extension. We show that the maximality of these curves implies that of certain ray class field extensions of each of the Deligne-Lusztig curves. Moreover, we show that the Giulietti-Korchm\'aros curve is equal to the above-mentioned ray class field extension of the Hermitian curve., Comment: 15 pages; various typos fixed

The smooth hermitian representations of a split reductive p-adic group whose restriction to a maximal hyperspecial compact subgroup contain a single K-type with Iwahori fixed vectors have been studied in [D. Barbasch, A. Moy, Classification of one K-type representations, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4245-4261] in the more general setting of modules for graded affine Hecke algebras with parameters. We show that every such one K-type module has nonzero Dirac cohomology (in the sense of [D. Barbasch, D. Ciubotaru, P. Trapa, The Dirac operator for graded affine Hecke algebras, arXiv:1006.3822]), and use Dirac operator techniques to determine the semisimple part of the Langlands parameter for these modules, thus completing their classification.

Let R ⟨ x ⟩ \mathbb R\langle x \rangle denote the ring of polynomials in g g freely noncommuting variables x = ( x 1 , … , x g ) x=(x_1,\dots ,x_g) . There is a natural involution ∗ * on R ⟨ x ⟩ \mathbb R\langle x \rangle determined by x j ∗ = x j x_j^*=x_j and ( p q ) ∗ = q ∗ p ∗ (pq)^*=q^* p^* , and a free polynomial p ∈ R ⟨ x ⟩ p\in \mathbb R\langle x \rangle is symmetric if it is invariant under this involution. If X = ( X 1 , … , X g ) X=(X_1,\dots ,X_g) is a g g tuple of symmetric n × n n\times n matrices, then the evaluation p ( X ) p(X) is naturally defined and further p ∗ ( X ) = p ( X ) ∗ p^*(X)=p(X)^* . In particular, if p p is symmetric, then p ( X ) ∗ = p ( X ) p(X)^*=p(X) . The main result of this article says if p p is symmetric, p ( 0 ) = 0 p(0)=0 and for each n n and each symmetric positive definite n × n n\times n matrix A A the set { X : A − p ( X ) ≻ 0 } \{X:A-p(X)\succ 0\} is convex, then p p has degree at most two and is itself convex, or − p -p is a hermitian sum of squares.

It is shown that, up to scaling and isometry, the only complete 4-manifold with an Einstein metric of positive sectional curvature which is also Hermitian with respect to some complex structure is the complex projective plane C P 2 \mathbb {CP}_2 , equipped with its Fubini-Study metric.

Let M M be a Riemannian manifold and let E → M E\to M be a Hermitian vector bundle with a Hermitian covariant derivative ∇ \nabla . Furthermore, let H ( 0 ) H(0) denote the Friedrichs extension of ∇ ∗ ∇ / 2 \nabla ^*\nabla /2 and let V : M → E n d ( E ) V:M\to \mathrm {End}(E) be a potential. We prove that if V V has a decomposition of the form V = V 1 − V 2 V=V_1-V_2 with V j ≥ 0 V_j\geq 0 , V 1 V_1 locally integrable and | V 2 | \left | V_2 \right | in the Kato class of M M , then one can define the form sum H ( V ) := H ( 0 ) ∔ V H(V):=H(0)\dotplus V in Γ L 2 ( M , E ) \Gamma _{\mathsf {L}^2}(M,E) without any further assumptions on M M . Applications to quantum physics are discussed.

We present a new proof and extension of the classical Sylvester Inertia Theorem to a pair of non-Hermitian matrices which satisfies the property that any real linear combination of the pair has only real eigenvalues. In the proof, we embed the given problem in a one-parameter family of related problems and examine the eigencurves of the family. The proof requires only elementary matrix theory and the Intermediate Value Theorem. The same technique is then used to extend Johnson’s extension of Sylvester’s Theorem on possible values of the inertia of a product of two matrices.

We consider the quadratic polynomials in three unitary generators, i.e. the elements of the group ∗ * -algebra of the free group with generators u 1 , u 2 , u 3 u_1, u_2, u_3 of the form f = ∑ j , k = 1 3 α j k u j ∗ u k f=\sum _{j, k=1}^{3}\alpha _{jk}u_{j}^{*}u_{k} , α j k ∈ C \alpha _{jk} \in \mathbb {C} . We prove that if f f is self-adjoint and T r ( f ( U 1 , U 2 , U 3 ) ) ≥ 0 \mathrm {Tr}(f(U_{1}, U_2 ,U_{3}))\ge 0 for arbitrary unitary matrices U 1 , U 2 , U 3 U_{1}, U_2, U_3 , then f f is a sum of hermitian squares. To prove this statement we reduce it to the question whether a certain Tarski sentence is true. Tarski’s decidability theorem thus provides an algorithm to answer this question. We use an algorithm due to Lazard and Rouillier for computing the number of real roots of a parametric system of polynomial equations and inequalities implemented in Maple to check that the Tarski sentence is true. As an application, we describe the set of parameters a 1 , a 2 , a 3 , a 4 a_1, a_2, a_3, a_4 such that there are unitary operators U 1 , … , U 4 U_1, \ldots , U_4 connected by the linear relation a 1 U 1 + a 2 U 2 + a 3 U 3 + a 4 U 4 = 0 a_1 U_1+a_2 U_2 +a_3 U_3 +a_4 U_4 =0 .

K. Ball has proved the “complex plank problem”: if ( x k ) k = 1 n \left (x_{k}\right )_{k=1}^{n} is a sequence of norm 1 1 vectors in a complex Hilbert space ( H , ⟨ ⋅ , ⋅ ⟩ ) \left (H, \, \langle \cdot ,\cdot \rangle \right ) , then there exists a unit vector x x for which \[ | ⟨ x , x k ⟩ | ≥ 1 / n , k = 1 , … , n . \left |\langle {x}, x_{k}\rangle \right |\geq 1/\sqrt {n}\,,\quad k=1, \ldots , n\,. \] In general, this result is not true on real Hilbert spaces. However, in special cases we prove that the same result holds true. In general, for some unit vector x x we have derived the estimate \[ | ⟨ x , x k ⟩ | ≥ max { λ 1 / n , 1 / λ n n } , \left |\langle {x}, x_{k}\rangle \right | \geq \max \left \{\sqrt {\lambda _{1}/n},\, 1/\sqrt {\lambda _{n}n}\right \}\,, \] where λ 1 \lambda _{1} is the smallest and λ n \lambda _{n} is the largest eigenvalue of the Hermitian matrix A = [ ⟨ x j , x k ⟩ ] A=\left [\langle {x_{j}}, x_{k}\rangle \right ] , j , k = 1 , … , n j, k=1, \ldots , n . We have also improved known estimates for the norms of homogeneous polynomials which are products of linear forms on real Hilbert spaces.

It is shown that over infinitely many imaginary quadratic fields there exists a Hermitian lattice in all even ranks n ≥ 2 n \ge 2 which is generated by its 4 n 4n minimal vectors but which is not generated by 2 n − 1 2n-1 minimal vectors.

In the present note we describe a family of H ∗ H^* -algebra structures on the set L 2 ( X ) L^2(X) of square integrable functions on a rank-one para-Hermitian symmetric space X X .

Consider the ring of all κ × κ \kappa \times \kappa column finite matrices over a ring R R . We prove that each such matrix is conjugate to a row and column finite matrix if and only if R R is right Noetherian and κ \kappa is countable. We then demonstrate that one can perform this conjugation on countably many matrices simultaneously. Some applications and limitations are given.

The gamma, beta and Dirichlet functions have been generalized in several ways by Ingham, Siegel, Bellman and Olkin. These authors defined them as integrals having the integrand as a scalar function of real symmetric matrix. In this article, we have defined and studied these functions when the integrand is a scalar function of Hermitian matrix.

In this paper we describe the general forms of all (nonlinear) continuous functionals on the sets of positive definite, positive semi-definite and Hermitian matrices which are multiplicative on the commuting elements. As a consequence, we obtain some new characterizations of the determinant on those classes of matrices.

In this paper we generalize the notion of p p -adic modular form to the Hermitian modular case and prove a formula that shows a coincidence between certain p p -adic Hermitian Eisenstein series and the genus theta series associated with Hermitian matrix with determinant p p .

Let G G be a finite group generated by unitary reflections in a Hermitian space V V , and let ζ \zeta be a root of unity. Let E E be a subspace of V V , maximal with respect to the property of being a ζ \zeta -eigenspace of an element of G G , and let C C be the parabolic subgroup of elements fixing E E pointwise. If χ \chi is any linear character of G G , we give a condition for the restriction of χ \chi to C C to be trivial in terms of the invariant theory of G G , and give a formula for the polynomial ∑ x ∈ G χ ( x ) T d ( x , ζ ) \sum _{x\in G}\chi (x)T^{d(x,\zeta )} , where d ( x , ζ ) d(x,\zeta ) is the dimension of the ζ \zeta -eigenspace of x x . Applications include criteria for regularity, and new connections between the invariant theory and the structure of G G .

We find full invariants for detecting non--zero elements in ${L}_0(F(t))\otimes \Q$, this group plays an important role in topology in the work done by Casson and Gordon., Comment: 8 pages

We obtain a non-Kahler almost Hermitian manifold of constant holomorphic sectional curvature by changing the almost complex structure in a Kahler manifold of constant holomorphic sectional curvature.

Two of the most useful inequality formulas for bounded linear operators on a Hilbert space are the Ldwner-Heinz and Reid's inequalities. The first inequality was generalized by Furuta (so called the Furuta inequality in the literature). We shall generalize the second one and obtain its related results. It is shown that these two generalized fundamental inequalities are all equivalent to one another. 1. NOTATIONS AND INTRODUCTION Throughout the paper we use capital letters to denote bounded linear operators acting on a Hilbert space H. T is positive (written T > 0 ) in case (Tx, x) > 0 for all x EH. If S and T are Hermitian, we write T > S in case T-S > O. T = U I T I is the polar decomposition of T with U the partial isometry, whereas I T I is the positive square root of the operator T*T, and U*U is the initial projection. Let I denote the identity operator. Under the polar decomposition of T we recall a well-known relation: I T* Iq= U I T Iq U* for q > 0 [1, p. 752]. The aim of this article is to define and characterize a generalized Reid inequality. Consequently, related inequalities and improved inequalities are given. Recall that if S > 0 and SK is Hermitian, then for x EH, I (SKx, x) I? 11 K 11 (Sx, x) is Reid's inequality [8], and the inequality was sharpened by Halmos [4, pp. 51, 244], where II K II is replaced by the spectral radius of K. Let a, ,3 E [0, 1] with a + ,3 > 1. For x,y EH let us call I (SK I SK Ic+,-1 x,y) I'll K IIc+/31 Sax SI y I an extended Reid's inequality. We shall call, for an obvious reason, the inequality (2) in Theorem 1 below a generalized Reid inequality, whereas (1) is the well-known altered Furuta inequality [2]. Our main result is characterizations of this inequality.

Given a certain kind of linear operator A (possibly a differential operator or a properly supported pseudodifferential operator) between sections of Hermitian vector bundles over a Riemannian manifold, a necessary and sufficient condition is obtained for the operator A to be solvable in a class of nonstandard sections in a generalized sense of weak solutions. The existence of a fundamental-solution-like internal section is established in the solvable case.

Let 0 < p, q, r E R be real numbers with p + 2r < (1 + 2r)q and 1 < q. Furuta (1987) proved that if bounded linear operators A, B E B(H) P+2r 1 on a Hilbert space H satisfy 0 < B < A, then B q < (BrAPBr) q. This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach *-algebra with continuous involution. Let A, B be bounded linear operators on a Hilbert space H. The celebrated Lbwner-Heinz inequality states the following; Theorem A (L6wner-Heinz inequality [4], [5]). Let A, B E B(H) satisfy 0 < B < A. If 0 < p < 1, then BP < AP. For an extension of Theorem A, Furuta obtained the following interesting inequality in [1] and one page elementary proof in [2]. Theorem B (Furuta inequality [1], [2]). Let 0 < p, q, r G R and A, B E B(H) satisfy 0 < B < A. If p + 2r < (1 + 2r)q and 1 < q as shown in the figure, then B q < (BrAPBr) q.

In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([i ] + 1)-reflexive, where [n ] denotes the greatest integer not n larger than '. By the result, we show that if 1?( ) = E Ai( )Bi is an 2=1 elementary operator on a C*-algebra A, then 'D is completely positive if and only if 'D is ([n1 ] + 1)-positive. In this paper, let H denote a complex Hilbert space. Let H(') denote the direct sum of n copies of H. For T E L(H), we write T(') for the operator on H(') which is the direct sum of n copies of T; the notation is extended to a subset of L(H) by S(n) = {T(n) E L(H(n)): T E S}. If S is a subspace of L(H), S is called n-reflexive if S(n) = ref (S(n)) =_ {T(n) E L(H(n)): T(n)X E [S(n)X], for all x E H (n)}, where [] denotes norm closed linear span. By the definiton, we have that if S is m-reflexive, then S is n-reflexive for n > m. A separating vector for a subspace S of L(H) is a vector x E H such that T 4 Tx, T E S, is an invective map. For x, y E H, let x 0 y denote the rank-one operator u | 4 (u, x)y. Let A denote a C*-algebra. Then A is called primitive, if A has a faithful irreducible representation on some Hilbert space. An elementary operator AP on A n is a linear mapping of the form AP: T F4 AiTBi, where {Ai},nL1 and {Bi},nU1 are i=l subsets of A. In this paper, we assume that all elementary operators are nonzero. A linear map 4J on A is positive (resp. hermitian-preserving) if 4)(T) is positive (resp. hermitian) for all positive (resp. hermitian) T in A. We define 4'n = 4{I 0 In: Mn(A) -4 Mn(A) by 4) 0 In((Tij)nxn) = (4)(Ti ))nxn. 4) is said to be n-positive if 4J 0 In is positive. If 4J is n-positive for all n, then 4J is said to be completely positive. In [4], Magajna states the following problem: For each positive integer r determine the smallest k = k(r) such that all rdimensional subspaces of L(H) are k-reflexive. In [4], Magajna proves k < r. In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([n] + 1)-reflexive. Also by this result, we study complete positivity of elementary operators on a C*-algebra A. We prove that if n 4 ) = Ai( )Bi is an elementary operator on a C*-algebra A, then 4) is i=l1 Received by the editors July 8, 1996 and, in revised form, May 14, 1997. 1991 Mathematics Subject Classification. Primary 47B47, 47B49; Secondary 46L05.

We will complete the list of universal binary Hermitian forms over imaginary quadratic fields by investigating three Hermitian forms missed by previous researchers.

We prove a convexity theorem on a generalized numerical range that combines and generalizes the following results: 1) Friedland and Loewy’s result on the existence of a nonzero matrix with multiple first eigenvalue in subspaces of hermitian matrices, 2) Bohnenblust’s result on joint positive definiteness of hermitian matrices, 3) the Toeplitz-Hausdorff Theorem on the convexity of the classical numerical range and its various generalizations by Au-Yeung, Berger, Brickman, Halmos, Poon, Tsing and Westwick.

The Weyl calculus is a means of constructing functions of a system of hermitian operators which do not necessarily commute with each other. This note gives a new proof of a formula, due to E. Nelson, for the Weyl calculus associated with a system of hermitian matrices.

We study the existence and uniqueness problems for Hermitian harmonic maps from Hermitian manifolds with boundary to Riemannian manifolds of nonpositive sectional curvature and with convex boundary. The complex analyticity of such maps and the related rigidity problems are also investigated.

Let H and C be n × n n \times n Hermitian matrices with C positive definite. Let H ( i 1 , … , i r ) H({i_1}, \ldots ,{i_r}) denote the submatrix of H formed by deleting the rows and columns i 1 , … , i r {i_1}, \ldots ,{i_r} , of H. In this paper, with r 1 + ⋯ + r k ≤ n {r_1} + \cdots + {r_k} \leq n , we study the roots of the determinantal equation det ( λ C − H ) = 0 \det (\lambda C - H) = 0 and those of \[ det ( ( λ C − H ) ( r 1 + ⋯ + r i − 1 + 1 , … , r 1 + ⋯ + r i ) ) = 0 \det ((\lambda C - H)({r_1} + \cdots + {r_{i - 1}} + 1, \ldots ,{r_1} + \cdots + {r_i})) = 0 \] for i = 1 , … , k i = 1, \ldots ,k .

We show that, as n -0oo , the product of the norm of the triangular truncation map on the n x n complex matrices with the distance from the normone hermitian n x n matrices to the nilpotents converges to 1/2. We also include an elementary proof of D. Herrero's characterization of the normal operators that are norm limits of nilpotents. Suppose n is a positive integer and let 'n, $n, An denote, respectively, the sets of all n x n complex matrices, strictly upper triangular n x n matrices, and nilpotent n x n matrices. There is a natural mapping Tn: A4 -+ $n, namely, Tn (T) replaces the entries on or below the main diagonal of T with zeroes. The map Tn is called triangular truncation on 4n . On an infinite-dimensional space, the triangular truncation mapping does not always yield the matrix of a bounded operator. This is related to the fact that the range of the mapping that sends a bounded harmonic function on the unit disk to its analytic part is not included in HOO . For example, if f(z) = log(l z), then u = 2i Im(f) is bounded in modulus by ir, but the analytic part of u, namely f, is not bounded. In terms of Toeplitz operators, Tu is an operator with norm 7r, but the upper triangular truncation of Tu is the formal matrix for Tf, which is not a bounded operator. The matrix for Tu is the matrix whose (i, i)-entry is 0 and (i, j)-entry is 1/(j i) for 1 < i $ j < oc. For each positive integer n, let Tu n be the n x n upper-left-hand corner of Tu, i.e., the (i, i)-entry of Tu,n is 0, and the (i, j)-entry of Tu,n is 1/(j i) for 1 < i $ j < n . It follows that IITu,nII < Xr foreach n, andthat II T(T ,n)IT n oc as n -oc. Hence I ITnIoc as n oc . Much work has been done in determining IITn . S. Kwapien and A. Pelczynski [KP, pp. 45-48] proved in 1970 that liTnil = O(log(n)), K. Davidson [D, p. 39] proved that 4 < liminfn 00 liTnil/log(n), and, in 1993, J. R. Angelos, C. Cowen, and S. K. Narayan [ACN] proved that lim l-Tnll/log(n) = 1/7r. n--*oo Received by the editors September 22, 1993; this paper was presented at a miniconference in honor of Eric Nordgren's sixtieth birthday held at the University of New Hampshire in June, 1993. 1991 Mathematics Subject Classification. Primary 47A58, 47B1 5; Secondary 15A60. ? 1995 American Mathematical Society 0002-9939/95 $1.00 + $.25 per page

In this note we shall get concrete expressions of hermitian operators on a closed subspace of C ( Ω ) C(\Omega ) which contains constant functions and separates points of Ω \Omega .

Using a criterion of Lawlor, it is shown that the cone over an isolated orbit of the adjoint action of SU ( n ) {\text {SU}}(n) on the unit sphere in the vector space of traceless n-by- n Hermitian symmetric matrices is area-minimizing for n > 2 n > 2 . Likewise, the cone over an isolated orbit of the adjoint action of SO ( n ) {\text {SO}}(n) is shown to be area-minimizing for n > 3 n > 3 .

We study the problem when a compact Hermitian ∗ {\ast } -Einstein surface M M is Kählerian and show that it is true if M M is additionally assumed to be either Einstein or anti-self-dual. We also prove that if the ∗ {\ast } -scalar curvature of M M is positive then M M is a conformally Kähler surface with positive first Chern class.

We study a class of hermitian maps on an algebra endowed with an indefinite inner product. We show that, in particular, the existence of a non-real eigenvalue is incompatible with the existence of a real eigenvalue having a right-invertible eigenvector. It also follows that for this class of maps the existence of an appropriate extremal for an indefinite Rayleigh quotient implies the nonexistence of nonreal eigenvalues. These results are intended to complement the Perron-Fröbenius and Kreĭn-Rutman theorems, and we conclude the paper by describing applications to ordinary and partial differential equations and to tridiagonal matrices.

We give a necessary and sufficient condition that a singular square matrix A A over an arbitrary field can be written as a product of two matrices with prescribed eigenvalues. Except when A A is a 2 × 2 2 \times 2 nonzero nilpotent, the condition is that the number of zeros among the eigenvalues of the factors is not less than the nullity of A A . We use this result to prove results about products of hermitian and positive semidefinite matrices simplifying and strengthening some known results.

We relate two different orbit decompositions of the flag variety. This allows us to pass from the closed formulas of Boe, Enright, and Shelton for the formal character of an irreducible highest weight representation to closed formulas for the distributional character written as a sum of characters of generalized principal series representations. Otherwise put, we give a dictionary between certain Lusztig-Vogan polynomials arising in Harish-Chandra module theory and the Kazhdan-Lusztig polynomials associated to a relative category O \mathcal {O} of Hermitian symmetric type.

We study anti-self-dual hermitian surfaces with odd first Betti number. We show that the twistor space of a Hopf surface M M has algebraic dimension 2, and we prove existence of half-conformally-flat metrics on the connected sum of copies of M M and C P 2 \mathbb {C}{\mathbb {P}_2} . Finally we emphasize some differences with the case b 1 {b_1} even.

Let H H be a positive semi-definite m n mn -by- m n mn Hermitian matrix, partitioned into m 2 {m^2} n n -square blocks H i j , i , j = 1 , … , m {H_{ij}},i,j = 1, \ldots ,m . We denote this by H = [ H i j ] H = [{H_{ij}}] . Consider the function f : M n → M r f:{M_n} \to {M_r} given by f ( X ) = X k f(X) = {X^k} (ordinary matrix product) and denote H f = [ f ( H i j ) ] {H_f} = [f({H_{ij}})] . We shall show that if H H is positive semi-definite then under some restrictions on H i j , H f {H_{ij}},{H_f} is also positive semi-definite. This generalizes familar results for Hadamard and ordinary products.

Equivalence of various norms on the unitization of a nonunital Banach algebra is established, with bounds ( 1 1 and 6 exp ⁡ ( 1 ) 6\exp (1) ) uniform over the class of such algebras. A tighter bound, 3 3 , is obtained in C ∗ {C^{\ast }} -algebras for elements with Hermitian nonunital parts.

We introduce a simple construction of a Siegel cusp form obtained by taking the difference between the Siegel Eisenstein series and the restricted Hermitian Eisenstein series. In addition, we present applications of the Siegel cusp form.

In this article, we classify all the Hermitian metrics on a complex product manifold M = X x Y with nonpositive holomorphic bisectional curvature. It is a generalization of a result by Zheng.

Let be a real continuous function on an interval, and consider the operator function defined for Hermitian operators . We will show that if is increasing w.r.t. the operator order, then for the operator function is convex. Let and be functions defined on an interval . Suppose is non-decreasing and is increasing. Then we will define the continuous kernel function by , which is a generalization of the Lowner kernel function. We will see that it is positive definite if and only if whenever for Hermitian operators , and we give a method to construct a large number of infinitely divisible kernel functions.

Let M n be the set of n x n complex matrices, and for every A ∈ M n , let Sp(A) denote the spectrum of A. For various types of products A 1 *... * A k on M n , it is shown that a mapping ∅: M n -> M n satisfying Sp(A 1 *... A k ) = Sp(O(A 1 ) *.... o(A k )) for all A 1,... A k 6 M n has the form X (S-1XS or A -> ξS- 1 X t S X - ξS -1 XS or A-> ξS- 1 X t S for some invertible S ∈ M n and scalar ξ. The result covers the special cases of the usual product A 1 *... * A k A 1 ... A k , the Jordan triple product A 1 * A 2 = A 1 * A 2 * A 1 , and the Jordan product A 1 * A 2 = (A 1 A 2 + A 2 A 1 )/2. Similar results are obtained for Hermitian matrices.

Let A A and B B be n × n n \times n matrices. We show that the matrix representing the linear transformation \[ X ↦ ( A X B + B X A ) T X \mapsto {(AXB + BXA)^T} \] (which is from the space of n × n n \times n matrices to itself) with respect to the usual basis is symmetric and show a similar symmetry property for the Fréchet derivative of a function f ( X ) = ∑ i = 0 ∞ a i X i f(X) = \sum \nolimits _{i = 0}^\infty {{a_i}{X^i}} defined on the space of n × n n \times n matrices.