Bulletin of Iranian Mathematical Society
Volume:41 Issue: 7, 2015

For an n-by-n complex matrix A in a block form with the (possibly) nonzero blocks only on the diagonal above the main one, we consider two other matrices whose nonzero entries are along the diagonal above the main one and consist of the norms or minimum moduli of the diagonal blocks of A. In this paper, we obtain two inequalities relating the numeical radii of these matrices and also determine when either of them becomes an equality.

Let P(λ) P(λ) be an n n-square complex matrix polynomial, and 1≤k≤n 1≤k≤n be a positive integer. In this paper, some algebraic and geometrical properties of the k k-numerical range of P(λ) P(λ) are investigated. In particular, the relationship between the k k-numerical range of P(λ) P(λ) and the k k-numerical range of its companion linearization is stated. Moreover, the k k-numerical range of the basic A A-factor block circulant matrix, which is the block companion matrix of the matrixpolynomial P(λ)=λ m I n −A P(λ)=λmIn−A, is studied.

Let m,ninmathbbN m,ninmathbbN, D D be a division ring, and M mtimesn (D) Mmtimesn(D) denote the bimodule of all mtimesn mtimesn matrices with entries from D D. First, we characterize one-sided submodules of M mtimesn (D) Mmtimesn(D) in terms of left row reduced echelon or right column reduced echelon matrices with entries from D D. Next, we introduce the notion of a nest module of matrices with entries from D D. We then characterize submodules of nest modules of matrices over D D in terms of certain finite sequences of left row reduced echelon or right column reduced echelon matrices with entries from D D. We use this result to characterize principal submodules of nest modules. We also describe subbimodules of nest modules of matrices. As a consequence, we characterize(one-sided) ideals of nest algebras of matrices over division rings.

Let φ(z)=z m, z∈U φ(z)=zm,z∈U, for some positive integer m m, and C φ Cφ be the composition operator on the Bergman space A 2 A2 induced by φ φ. In this article, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators C ∗ φ C φ, C φ C ∗ φ Cφ∗Cφ,CφCφ∗ as well as self-commutator and anti-self-commutators of C φ Cφ. We also find the eigenfunctions of these operators.

For A,B∈M nm, A,B∈Mnm, we say that A A is left matrix majorized (resp. left matrix submajorized) by B B and write A≺ ℓ B A≺ℓB (resp. A≺ ℓs B A≺ℓsB), if A=RB A=RB for some n×n n×n row stochastic (resp. row substochastic) matrix R. R. Moreover, we define the relation ∼ ℓs ∼ℓs on M nm Mnm as follows: A∼ ℓs B A∼ℓsB if A≺ ℓs B≺ ℓs A. A≺ℓsB≺ℓsA. This paper characterizes all linear preservers and all linear strong preservers of ≺ ℓs ≺ℓs and ∼ ℓs ∼ℓs from M nm Mnm to M nm Mnm.

Let A A and B B be C ∗ ∗ -algebras. Assume that A A is of real rank zero and unital with unit I I and k>0 k>0 is a real number. It is shown that if Φ:A→B Φ:A→B is an additive map preserving |⋅| k |⋅|k for all normal elements; that is, Φ(|A| k)=|Φ(A)| k Φ(|A|k)=|Φ(A)|k for all normal elements A∈A A∈A, Φ(I) Φ(I) is a projection, and there exists a positive number c c such that Φ(iI)Φ(iI) ∗ ≤cΦ(I)Φ(I) ∗ Φ(iI)Φ(iI)∗≤cΦ(I)Φ(I)∗, then Φ Φ is the sum of a linear Jordan *-homomorphism and a conjugate-linear Jordan * homomorphism. If, moreover, the map Φ Φ commutes with |.| k |.|k on A A, then Φ Φ is the sum of a linear *-homomorphism and a conjugate-linear *-homomorphism. In the case when k≠1 k≠1, the assumption Φ(I) Φ(I) being a projection can be deleted.

This paper presents a proof of Stirling's formula using Haar wavelets and some properties of Hilbert space, such as Parseval's identity. The present paper shows a connection between Haar wavelets and certain sequences.

Let A A and B B be two C ∗ C∗ -algebras such that B B is prime. In this paper, we investigate the additivity of maps Φ Φ from A A onto B B that are bijective, unital and satisfy Φ(AP+ηPA ∗)=Φ(A)Φ(P)+ηΦ(P)Φ(A) ∗, Φ(AP+ηPA∗)=Φ(A)Φ(P)+ηΦ(P)Φ(A)∗, for all A∈A A∈A and P∈{P 1, I A −P 1} P∈{P1,IA−P1} where P 1 P1 is a nontrivial projection in A A. If η η is a non-zero complex number such that |η|≠1 |η|≠1, then Φ Φ is additive. Moreover, if η η is rational<,> then Φ Φ is ∗ ∗ -additive.

Suppose π:A→B π:A→B is a surjective unital ∗ ∗ -homomorphism between C*-algebras A A and B B, and 0≤a≤1 0≤a≤1 with a∈A a∈A. We give a sufficient condition that ensures there is a proection p∈A p∈A such that π(p)=π(a) π(p)=π(a). An easy consequence is a result of [L. G. Brown and G. k. Pedersen, C*-algebras of realrank zero, \textit{J. Funct. Anal.} {99} (1991) 131--149] that such a p p exists when A A has real rank zero.

We introduce a matricial Toeplitz transform and prove that the Toeplitz transform of a second order recurrence sequence is another second order recurrence sequence. We investigate the injectivity of this transform and show how this distinguishes the Fibonacci sequence among other recurrence sequences. We then obtain new Fibonacci identities as an application of our transform.

The main result of this article is that for collections of entry-wise non-negative matrices the property of possessing a standard triangularization is stable under approximation. The methodology introduced to prove this result allows us to offer quick proofs of the corresponding results of [B. R. Yahaghi, Near triangularizability implies triangularizability, Canad. Math. Bull. 47, (2004), no. 2, 298--313], and [A. A. Jafarian, H. Radjavi, P. Rosenthal and A. R. Sourour, Simultaneous, triangularizability, near commutativity and Rota's theorem, Trans. Amer. Math. Soc. 347, (1995), no. 6, 2191--2199] on the approximations and triangularizability of collections of operators and matrices. In conclusion we introduce and explore a related topology on the power sets of metric spaces.

In his paper mentioned in the title, which appears in the same issue of this journal, MehdiRadjabalipour derives the cyclic decomposition of an algebraic linear transformation. A more general structure theory for linear transformations appears in Irving Kaplansky's lovely 1954 book on infinite abelian groups. We present a translation of Kaplansky's results for abelian groups into the terminology of linear transformations. We add an additional translation of a ring-theoretic result to give a characterization of algebraicallyhyporeflexive transformations and the strict closure of the set of polynomials in a transformation T T.

The famous primary and cyclic decomposition theorems along with the tightly related rational and Jordan canonical forms are extended to linear spaces of infinite dimensions with counterexamples showing the scope of extensions.

The aim of this note is to study the submajorization inequalities for τ τ-measurable operators in a semi-finite von Neumann algebra on a Hilbertspacewithanormal faithful semi-finite trace τ τ. The submajorization inequalities generalize some results duetoZhang,FuruichiandLin,etc..

One of unsolved problems in quantum measurement theory is to characterize coexistence of quantum effects. In this paper, applying positive operator matrix theory, we give a mathematical characterization of the witness set of coexistence of quantum effects and obtain a series of properties of coexistence. We also devote to characterizing bijective morphisms on quantum effects leaving the witness set invariant. Furthermore, applying linear maps preserving commutativity, which are characterized by Choi, Jafarian and Radjavi [Linear maps preserving commutativity, Linear Algebra Appl. 87 (1987), 227--241.], we classify multiplicative general morphisms leaving the witness set invariant on finite dimensional Hilbert space effect algebras.