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.     

On the Schur multiplier norm of matrices

The Schur product of two n×n complex matrices A=(a_ij), B=(b_ij) is defined by A°B=(a_ijb_ij. By a result of Schur [2], the algebra of n×n matrices with Schur product and the usual addition is a commutative Banach algebra under the operator norm (the norm of the operator defined on $\text{C}$ⁿ by the matrix). For a fixed matrix A, the norm of the operator B?A°B on this Banach algebra is called the Schur multiplier norm of A, and is denoted by ∥A∥^m. It is proved here that $\text{∥A∥=∥U}{}^1\text{AU∥}{}_{\mathrm{m}}$ for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥A∥_m=∥A∥ iff there exist two permutations P, Q, a p×p (1?p?n) unitary U, an (n?p)×(n?p)1 contraction C, and a nonnegative number λ such that

$\text{A=λP}\text{U}\text{0}\text{0}\text{C}\text{Q;}$

and this is so iff $\text{∥A°}\text{A}\text{?}\text{∥=∥A∥}{}^2$, where ā is the matrix obtained by taking entrywise conjugates of A.     

Schur multiplier of a group of elementary matrices of finite order

If a ring is finitely generated as a module over its center, then for the Schur multiplier coincides with. For a representation of is obtained which is analogous to van der Kallen's representation for a commutative ring.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 86, pp. 162–169, 1979.     

An infinite class of Williamson matrices

Only a finite number of Hadamard matrices of Williamson type are known so far; it has been conjectured that one such exists of any order 4t. An infinite family is constructed here, and as a corollary it is shown that an Hadamard matrix of order 6(q + 1) exists if q is a prime power ≡ 1 (mod 4).     

Interpolation of Schur multiplier spaces

Let denote the space of all Schur mulptiliers on the Schatten class . It is proved that there is a linear map which is a contraction on both and but which is not bounded on for all . Consequently, for any , is not an interpolation space between and . Received April 3, 1998 / in final form September 22, 1999 / Published online July 20, 2000     

On the norm of an idempotent Schur multiplier on the Schatten class

We show that if the norm of an idempotent Schur multiplier on the Schatten class lies sufficiently close to , then it is necessarily equal to . We also give a simple characterization of those idempotent Schur multipliers on whose norm is .

     

Multiplicative Lie algebras and Schur multiplier

In this paper, we attempt to study the structure of multiplicative Lie algebras, the theory of extensions, the second cohomology groups of multiplicative Lie algebras, and in turn the Schur multipliers. The Schur–Hopf formula is established for multiplicative Lie algebras. We also introduce the group of nontrivial relations satisfied by the Lie product in a multiplicative Lie algebra, and study it as a functor arising from the presentations of multiplicative Lie algebras. Some applications in K-theory are also discussed.     

On the Schur multiplier of augmented algebras

Letk be a field, andA a finitely generatedk-algebra, with augmentation. Suppose there is a presentation ofA 0→I→R→A→0 whereR is a finitely generated freek-algebra andI is non-zero. IfA is infinite dimensional overk, Lewin proved thatR/I ² is not finitely presented. A stronger statement would be that the ‘Schur multiplier’ ofR/I ² is not finite dimensional. In the case thatA is an augmented domain, we prove this stronger statement, and some related statements.     

The Schur multiplier of central product of groups

Let G be a central product of two groups H and K. We study second cohomology group of G, having coefficients in a divisible abelian group D with trivial G-action, in terms of the second cohomology groups of certain quotients of H and K. In particular, for $D={\mathbb{C}}^{?}$, some of our results provide a refinement of results from Wiegold (1971) [10] and Eckmann et al. (1973) [2].     

On the Schur multiplier of special p-groups

Let G be a special p-group minimally generated by $d\ge 3$ elements and having derived subgroup of order $p^{\frac{1}{2}d(d?1)}$. Berkovich asked to find the Schur multiplier and covering groups of such groups G Berkovich and Janko (2011) [1]. We try to give an answer to this question in this article.     

The Schur multiplier of p-groups with large derived subgroup

For any given p-group of order p ⁿ (n ≥ 4) with derived subgroup of order p ^n-2 we will show that the order of its Schur multiplier is less than |G ^'|/2 when p = 2 and |G ^'| in the other cases.     

Invariant factors and generalized Schur matrices

The invariant factors of a generalization of the Schur matrix are found.     

A Schur decomposition for Hamiltonian matrices

A Schur-type decomposition for Hamiltonian matrices is given that relies on unitary symplectic similarity transformations. These transformations preserve the Hamiltonian structure and are numerically stable, making them ideal for analysis and computation. Using this decomposition and a special singular-value decomposition for unitary symplectic matrices, a canonical reduction of the algebraic Riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution. After presenting some real decompositions for real Hamiltonian matrices, we look into the possibility of an orthogonal symplectic version of the QR algorithm suitable for Hamiltonian matrices. A finite-step initial reduction to a Hessenberg-type canonical form is presented. However, no extension of the Francis implicit-shift technique was found, and reasons for the difficulty are given.     

Decomposition of infinite matrices

We consider the problem of decomposing a matrix of integers according to constraints on the row and column sums, in the case when the original matrix is infinite. Some necessary conditions and some sufficient conditions are found for the existence of such a decomposition, generalizing results of Ball for the finite case. Connections with the Transversal Problem for infinite sets are pointed out.     

On dimension of the Schur multiplier of nilpotent Lie algebras

Let L be an n-dimensional non-abelian nilpotent Lie algebra and $ s(L) = \frac{1} {2}(n - 1)(n - 2) + 1 - \dim M(L) $ s(L) = \frac{1} {2}(n - 1)(n - 2) + 1 - \dim M(L) where M(L) is the Schur multiplier of L. In [Niroomand P., Russo F., A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra (in press)] it has been shown that s(L) ≥ 0 and the structure of all nilpotent Lie algebras has been determined when s(L) = 0. In the present paper, we will characterize all finite dimensional nilpotent Lie algebras with s(L) = 1; 2.