On the self-inverse operators

Let T be an operator on a Hilbert space . The problem of computing of the norm of T, norm of selfcommutators of T, and the numerical radius of T are discussed in many papers and a number of textbooks. In this paper we determine the relationships between these values for self inverse operators and explain how we can determine any three of these (‖T‖, ‖[T*,T]‖, ‖{T*,T}‖, and the numerical radius of T) by knowing any one of them. Also, we find the spectrum of T*T,[T*,T] and {T*,T} in the case that T is self inverse and the spectrum of T*T is an interval. Finally, by giving some examples on automorphic composition operators, we show that these results make it possible to replace lengthy computation with quick ones. Research partially supported by the Shiraz university Research Council Grant No. 86-GR-SC-32.     

Linear maps between C*-algebras that are *-homomorphisms at a fixed point

Let A and B be C*-algebras. A linear map T : A → B is said to be a *-homomorphism at an element z ∈ A if ab* = z in A implies T (ab*) = T (a)T (b)* = T (z), and c*d = z in A gives T (c*d) = T (c)*T (d) = T (z). Assuming that A is unital, we prove that every linear map T : A → B which is a *-homomorphism at the unit of A is a Jordan *-homomorphism. If A is simple and infinite, then we establish that a linear map T : A → B is a *-homomorphism if and only if T is a *-homomorphism at the unit of A. For a general unital C*-algebra A and a linear map T : A → B, we prove that T is a *-homomorphism if, and only if, T is a *-homomorphism at 0 and at 1. Actually if p is a non-zero projection in A, and T is a ^?-homomorphism at p and at 1 ? p, then we prove that T is a Jordan *-homomorphism. We also study bounded linear maps that are *-homomorphisms at a unitary element in A.     

Weyl’s Theorem for Algebraically Quasi-class A Operators

If T or T* is an algebraically quasi-class A operator acting on an infinite dimensional separable Hilbert space then we prove that Weyl’s theorem holds for f(T) for every f ∈ H(σ(T)), where H(σ(T)) denotes the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is algebraically quasi-class A then a-Weyl’s theorem holds for f(T). Also, if T or T* is an algebraically quasi-class A operator then we establish that the spectral mapping theorems for the Weyl spectrum and the essential approximate point spectrum of T for every f ∈ H(σ(T)), respectively. This research was supported by the Kyung Hee University Research Fund in 2007 (KHU- 20071605).     

Reconstructing trees from two cards

Let ?? be the class of unlabeled trees. An unlabeled vertex‐deleted subgraph of a tree T is called a card. A collection of cards is called a deck. We say that the tree T has a deck D if each card in D can be obtained by deleting distinct vertices of T. If T is the only unlabeled tree that has the deck D, we say that T is ??‐reconstructible from D. We want to know how large of a deck D is necessary for T to be ??‐reconstructible. We define ??rn(T) as the minimum number of cards in a deck D such that T is ??‐reconstructible from D. It is known that ??rn(T)≤3, but it is conjectured that ??rn(T)≤2 for all trees T. We prove that the conjecture holds for all homeomorphically irreducible trees. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 243–257, 2010     

Ordering Trees with Nearly Perfect Matchings by Algebraic Connectivity

Let T2k＋1 be the set of trees on 2k＋1 vertices with nearly perfect matchings and α（T） be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T2k＋1. Specifically, 10 trees T2,T3,... ,T11 and two classes of trees T（1） and T（12） in T2k＋1 are introduced. It is shown in this paper that for each tree T^′1,T^″1∈T（1）and T^′12,T^″12∈T（12） and each i,j with 2≤i〈j≤11,α（T^′1）=α（T^″1）〉α（Tj）〉α（T^′12）=α（T^″12）.It is also shown that for each tree T with T∈T2k＋1／（T（1）∪{T2,T3,…,T11}∪T（12））,α（T^′12）〉α（T）.     

Relative T -Injective Modules and Relative T -Flat Modules

Let T be a Wakamatsu tilting module. A module M is called (n, T)-copure injective (resp. (n, T)-copure flat) if ɛ _T ¹ (N, M) = 0 (resp. Γ₁ ^T (N, M) = 0) for any module N with T-injective dimension at most n (see Definition 2.2). In this paper, it is shown that M is (n, T)-copure injective if and only if M is the kernel of an I _n (T)-precover f: A → B with A ∈ Prod T. Also, some results on Prod T-syzygies are presented. For instance, it is shown that every nth Prod T-syzygy of every module, generated by T, is (n, T)-copure injective.     

A Note on the Browder＇s and Weyl＇s Theorem

Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl＇s theorem for operator satisfying E（T） = π（T）. An application is also given.     

Joining-rank and the structure of finite rank mixing transformations

A new isomorphism invariant of zero-entropy maps, calledjoiningrank, is presented. Written jrk(T), it is a value in N ∪ {∞}. The depth of factors ofT, and the size of its essential commutant EC(T), are upper bounded by jrk(T). IfT is mixing then jrk(T)≦rank(T). ForT with finite joining rank, we obtain a structure theorem for the commutant group ofT: it is a certain twisted product of Z with EC(T). As forT itself, it must be anm-point extension of thenth power of a prime transformationS having trivial commutant. Also, jrk(T)=m·n·jrk(S). Thecovering-number, k(T), is a number in [0, 1] obeying 1/k(T)≦rank(T). Letting α(T)∈[0, 1] denoteT’s partial mixing, jrk(T) is dominated by 1/[k(T)+α(T)-1]. In particular, a rank-1T with partial mixing exceeding 1/2 has minimal self-joinings. Combined with Kalikow’s deep theorem that, forT rank one, 2-fold mixing implies mixing of all orders, our technique yields that a mixing suchT has minimal self-joinings of all orders. ThusT may be used as the seed for Rudolph's counterexample machine. Research partially supported by NSF grant DMS 8501519.     

On the Weyl Spectrum: Spectral Mapping Theorem and Weyl's Theorem

It is shown that ifTis a dominant operator or an analytic quasi-hyponormal operator on a complex Hilbert space and iffis a function analytic on a neighborhood of σ(T), then σ_w(f(T)) = f(σ_w(T)), where σ(T) and σ_w(T) stand respectively for the spectrum and the Weyl spectrum ofT; moreover, Weyl's theorem holds forf(T) + Fif “dominant” is replaced by “M-hyponormal,” whereFis any finite rank operator commuting withT. These generalize earlier results for hyponormal operators. It is also shown that there exist an operatorTand a finite rank operatorFcommuting withTsuch that Weyl's theorem holds forTbut not forT + F. This answers negatively a problem raised by K. K. Oberai (Illinois J. Math.21, 1977, 84–90). However, ifTis required to be isoloid, then the statement that Weyl's theorem holds forTwill imply it holds forT + F.     

A note on injectiveC(T)-spaces and the Amir boundary

We sovle in the negative a problem of Wolfe ifC(T _A ) is an injective Banach space wheneverC(T) is injective,T compact, andT _A is the Amir boundary ofT (i.e., the complement of the maximal open extremally disconnected subset ofT). In particular, we findT such thatC(T) is aP ₃-space andT _A ∼βN\N. The author’s research was partially supported by a grant of MEN, Poland.     

Products of Toeplitz Operators on the Polydisk

This paper studies products of Toeplitz operators on the Hardy space of the polydisk. We show that T _f T _g = 0 if and only if T _f T _g is a finite rank if and only if T _f or T _g is zero. The product T _f T _g is still a Toeplitz operator if and only if there is a h $ \in $ L $ \infty $ (T ⁿ ) such that T _f T _g - T _h is a finite rank operator. We also show that there are no compact simi-commutators with symbols pluriharmonic on the polydisk. Submitted: October 5, 2000     

On the Existence of Towers in Pseudo-Tree Algebras

A tower in a Boolean algebra (BA) is a strictly increasing sequence, of regular order type, of elements of the algebra different from 1 but with sum 1. A pseudo-tree is a partially ordered set T such that the set T↓t = {s ∈ T:s < t} is linearly ordered for every t ∈ T. If that set is well-ordered, then T is a tree. For any pseudo-tree T, the BA Treealg(T) is the algebra of subsets of T generated by all of the sets T↑t = {s ∈ T:t ≤ s}. The main theorem of this note is a characterization in tree terms of when Treealg(T) has a tower of order type κ (given in advance).     

On the Property (gw)

In this note we introduce and study the property (gw), which extends property (w) introduced by Rakoc̆evic in [23]. We investigate the property (gw) in connection with Weyl type theorems. We show that if T is a bounded linear operator T acting on a Banach space X, then property (gw) holds for T if and only if property (w) holds for T and Π _a (T) = E(T), where Π _a (T) is the set of left poles of T and E(T) is the set of isolated eigenvalues of T. We also study the property (gw) for operators satisfying the single valued extension property (SVEP). Classes of operators are considered as illustrating examples. The second author was supported by Protars D11/16 and PGR- UMP.     

The iterated Aluthge transforms of a matrix converge

Given an r×r complex matrix T, if T=U|T| is the polar decomposition of T, then, the Aluthge transform is defined byΔ(T)=|T|^1/2U|T|^1/2. Let Δn(T) denote the n-times iterated Aluthge transform of T, i.e., Δ⁰(T)=T and Δn(T)=Δ(Δn−1(T)), n∈N. We prove that the sequence {Δn(T)}_n∈N converges for every r×r matrix T. This result was conjectured by Jung, Ko and Pearcy in 2003. We also analyze the regularity of the limit function.     

Lifting trees of prime ideals to bezout extension domains

We prove that if an extension R ? T of commutative rings satisfies the going-up property, then any tree of prime ideals of R with at most two branches or in which each branch has finite length is covered by some corresponding tree of prime ideals of T. In particular, if R ? T is an integral extension and R is Noetherian, then each tree in Spec(R) can be covered by a tree in Spec(T). We also prove that if R is an integral domain, then each tree T in Spec(/2) can be covered by a tree in Spec(T) for some Bezout domain T containing R. If T has only finitely many branches, it can further be arranged that the Bezout domain T be an overring of R. However, in general, it cannot be arranged that T be covered from a Prüfer overring of R, thus answering negatively a question of D D. Anderson.     

Weyl’s theorem for totally hereditarily normaloid operators

A Banach space operatorT ∈B(χ) is said to behereditarily normaloid, denotedT ∈ ℋN, if every part ofT is normaloid;T ∈ ℋN istotally hereditarily normaloid, denotedT ∈ ℑHN, if every invertible part ofT is also normaloid. Class ℑHN is large; it contains a number of the commonly considered classes of operators. The operatorT isalgebraically totally hereditarily normaloid, denotedT ∈a — ℑHN, both non-constant polynomialp such thatp(T) ∈ ℑHN. For operatorsT ∈a − ℑHN, bothT andT* satisfy Weyl’s theorem; if also either ind(T−μ)≥0 or ind(T−μ)≤0 for all complexμ such thatT−μ is Fredholm, thenf(T) andf(T*) satisfy Weyl’s theorem for all analytic functionsf ∈ ℋ(σ(T)). For operatorsT ∈a — ℑHN such thatT has SVEP,T* satisfiesa-Weyl’s theorem.     

Almost periodic measures on the torus

Given a skew product flow (T,T ²) on the two torus, we construct a family of flows onT ³ parametrized by elements of the circleT. We show that under a certain condition on (T,T ²) almost every flow in this family is strictly ergodic. This is used to characterize minimal subsets of the flow (T,P(T ²)) induced byT on the space of probability measures onT ². Using a result of M. Herman, we give an example to show that this characterization does not hold for everyT. To the memory of Shlomo Horowitz     

On the radicals of algebras generated by a contraction

Let T be a contraction on a Banach space, and let A _T (W _T) be the closure in the uniform (respectively, in the weak) operator topology of the set of all polynomials in T. The asymptotic behavior of the radicals of the algebras A _T and W _T is studied.     

On a Field-Theoretic Invariant for Extensions of Commutative Rings

Abstract

If R ? T is an extension of (commutative integral) domains, Λ(T/R) is defined as the supremum of the lengths of chains of intermediate fields in the extension k _R (Q ∩ R) ? k _T (Q), where Q runs over the prime ideals of T. The invariant Λ(T/R) is determined in case R and T are adjacent rings and in case Spec(R) = Spec(T) as sets. It is proved that if R is a domain with integral closure R′, then Λ(T/R) = 0 for all overrings T of R if and only if R′ is a Prüfer domain such that Λ(R′/R) = 0. If R ? T are domains such that the canonical map Spec(T) → Spec(R) is a homeomorphism (in the Zariski topology), then Λ(T/R) is bounded above by the supremum of the lengths of chains of rings intermediate between R and T. Examples are given to illustrate the sharpness of the results.     

Reconstructible and Half-Reconstructible Tournaments: Application to Their Groups of Hemimorphisms

Let T and T¹ be tournaments with n elements, E a basis for T, E′ a basis for T′, and k ≥ 3 an integer. The dual of T is the tournament T” of basis E defined by T(x, y) = T(y, x) for all x, y ε E. A hemimorphism from T onto T′ is an isomorphism from T onto T” or onto T. A k-hemimorphism from T onto T′ is a bijection f from E to E′ such that for any subset X of E of order k the restrictions T/X and T¹/f(X) are hemimorphic. The set of hemimorphisms of T onto itself has group structure, this group is called the group of hemimorphisms of T. In this work, we study the restrictions to n – 2 elements of a tournament with n elements. In particular, we prove: Let k ≥ 3 be an integer, T a tournament with n elements, where n ≥ k + 5. Then the following statements are equivalent: (i) All restrictions of T to subsets with n – 2 elements are k-hemimorphic. (ii) All restrictions of T to subsets with n – 2 elements are 3-hemimorphic. (iii) All restrictions of T to subsets with n – 2 elements are hemimorphic. (iv) All restrictions of T to subsets with n – 2 elements are isomorphic, (v) Either T is a strict total order, or the group of hemimorphisms of T is 2-homogeneous.