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26. CJM 2009 (vol 62 pp. 242)

Azagra, Daniel; Fry, Robb
 A Second Order Smooth Variational Principle on Riemannian Manifolds We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature. Keywords:smooth variational principle, Riemannian manifoldCategories:58E30, 49J52, 46T05, 47J30, 58B20

27. CJM 2009 (vol 62 pp. 133)

Makarov, Konstantin A.; Skripka, Anna
 Some Applications of the Perturbation Determinant in Finite von Neumann Algebras In the finite von Neumann algebra setting, we introduce the concept of a perturbation determinant associated with a pair of self-adjoint elements $H_0$ and $H$ in the algebra and relate it to the concept of the de la Harpe--Skandalis homotopy invariant determinant associated with piecewise $C^1$-paths of operators joining $H_0$ and $H$. We obtain an analog of Krein's formula that relates the perturbation determinant and the spectral shift function and, based on this relation, we derive subsequently (i) the Birman--Solomyak formula for a general non-linear perturbation, (ii) a universality of a spectral averaging, and (iii) a generalization of the Dixmier--Fuglede--Kadison differentiation formula. Keywords:perturbation determinant, trace formulae, von Neumann algebrasCategories:47A55, 47C15, 47A53

28. CJM 2009 (vol 61 pp. 1239)

Davidson, Kenneth R.; Yang, Dilian
 Periodicity in Rank 2 Graph Algebras Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of $\mathrm{C}^*(\mathbb{F}^+_{\theta})$. The periodic $\mathrm{C}^*$-algebras are characterized, and it is shown that $\mathrm{C}^*(\mathbb{F}^+_{\theta}) \simeq \mathrm{C}(\mathbb{T})\otimes\mathfrak{A}$ where $\mathfrak{A}$ is a simple $\mathrm{C}^*$-algebra. Keywords:higher rank graph, aperiodicity condition, simple $\mathrm{C}^*$-algebra, expectationCategories:47L55, 47L30, 47L75, 46L05

29. CJM 2009 (vol 61 pp. 282)

Bouya, Brahim
 Closed Ideals in Some Algebras of Analytic Functions We obtain a complete description of closed ideals of the algebra $\cD\cap \cL$, $0<\alpha\leq\frac{1}{2}$, where $\cD$ is the Dirichlet space and $\cL$ is the algebra of analytic functions satisfying the Lipschitz condition of order $\alpha$. Categories:46E20, 30H05, 47A15

30. CJM 2009 (vol 61 pp. 241)

Azamov, N. A.; Carey, A. L.; Dodds, P. G.; Sukochev, F. A.
 Operator Integrals, Spectral Shift, and Spectral Flow We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general \vNa s. For semifinite \vNa s we give applications to the Fr\'echet differentiation of operator functions that sharpen existing results, and establish the Birman--Solomyak representation of the spectral shift function of M.\,G.\,Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow. Categories:47A56, 47B49, 47A55, 46L51

31. CJM 2009 (vol 61 pp. 190)

Lu, Yufeng; Shang, Shuxia
 Bounded Hankel Products on the Bergman Space of the Polydisk We consider the problem of determining for which square integrable functions $f$ and $g$ on the polydisk the densely defined Hankel product $H_{f}H_g^\ast$ is bounded on the Bergman space of the polydisk. Furthermore, we obtain similar results for the mixed Haplitz products $H_{g}T_{\bar{f}}$ and $T_{f}H_{g}^{*}$, where $f$ and $g$ are square integrable on the polydisk and $f$ is analytic. Keywords:Toeplitz operator, Hankel operator, Haplitz products, Bergman space, polydiskCategories:47B35, 47B47

32. CJM 2009 (vol 61 pp. 50)

Chen, Huaihui; Gauthier, Paul
 Composition operators on $\mu$-Bloch spaces Given a positive continuous function $\mu$ on the interval $0 Categories:47B33, 32A70, 46E15 33. CJM 2008 (vol 60 pp. 1010) Galé, José E.; Miana, Pedro J. $H^\infty$Functional Calculus and Mikhlin-Type Multiplier Conditions Let$T$be a sectorial operator. It is known that the existence of a bounded (suitably scaled)$H^\infty$calculus for$T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra$\Lambda_{\infty,1}^\alpha(\R^+)$. Such an algebra includes functions defined by Mikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for$T$. In this paper, we use fractional derivation to analyse in detail the relationship between$\Lambda_{\infty,1}^\alpha$and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence. Keywords:functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliersCategories:47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22 34. CJM 2008 (vol 60 pp. 758) Bercovici, H.; Foias, C.; Pearcy, C.  On the Hyperinvariant Subspace Problem. IV This paper is a continuation of three recent articles concerning the structure of hyperinvariant subspace lattices of operators on a (separable, infinite dimensional) Hilbert space$\mathcal{H}$. We show herein, in particular, that there exists a universal'' fixed block-diagonal operator$B$on$\mathcal{H}$such that if$\varepsilon>0$is given and$T$is an arbitrary nonalgebraic operator on$\mathcal{H}$, then there exists a compact operator$K$of norm less than$\varepsilon$such that (i)$\Hlat(T)$is isomorphic as a complete lattice to$\Hlat(B+K)$and (ii)$B+K$is a quasidiagonal,$C_{00}$, (BCP)-operator with spectrum and left essential spectrum the unit disc. In the last four sections of the paper, we investigate the possible structures of the hyperlattice of an arbitrary algebraic operator. Contrary to existing conjectures,$\Hlat(T)$need not be generated by the ranges and kernels of the powers of$T$in the nilpotent case. In fact, this lattice can be infinite. Category:47A15 35. CJM 2008 (vol 60 pp. 520) Chen, Chang-Pao; Huang, Hao-Wei; Shen, Chun-Yen  Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences Let$A=(a_{j,k})_{j,k \ge 1}$be a non-negative matrix. In this paper, we characterize those$A$for which$\|A\|_{E, F}$are determined by their actions on decreasing sequences, where$E$and$F$are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces:$\ell_p$,$d(w,p)$, and$\ell_p(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour. Keywords:norms of matrices, normed Riesz spaces, weighted mean matrices, NÃ¶rlund mean matrices, summability matrices, matrices with row decreasingCategories:15A60, 40G05, 47A30, 47B37, 46B42 36. CJM 2007 (vol 59 pp. 1207) Bu, Shangquan; Le, Christian $H^p$-Maximal Regularity and Operator Valued Multipliers on Hardy Spaces We consider maximal regularity in the$H^p$sense for the Cauchy problem$u'(t) + Au(t) = f(t)\ (t\in \R)$, where$A$is a closed operator on a Banach space$X$and$f$is an$X$-valued function defined on$\R$. We prove that if$X$is an AUMD Banach space, then$A$satisfies$H^p$-maximal regularity if and only if$A$is Rademacher sectorial of type$<\frac{\pi}{2}$. Moreover we find an operator$A$with$H^p$-maximal regularity that does not have the classical$L^p$-maximal regularity. We prove a related Mikhlin type theorem for operator valued Fourier multipliers on Hardy spaces$H^p(\R;X)$, in the case when$X$is an AUMD Banach space. Keywords:$L^p$-maximal regularity,$H^p$-maximal regularity, Rademacher boundednessCategories:42B30, 47D06 37. CJM 2007 (vol 59 pp. 966) Forrest, Brian E.; Runde, Volker; Spronk, Nico  Operator Amenability of the Fourier Algebra in the$\cb$-Multiplier Norm Let$G$be a locally compact group, and let$A_{\cb}(G)$denote the closure of$A(G)$, the Fourier algebra of$G$, in the space of completely bounded multipliers of$A(G)$. If$G$is a weakly amenable, discrete group such that$\cstar(G)$is residually finite-dimensional, we show that$A_{\cb}(G)$is operator amenable. In particular,$A_{\cb}(\free_2)$is operator amenable even though$\free_2$, the free group in two generators, is not an amenable group. Moreover, we show that if$G$is a discrete group such that$A_{\cb}(G)$is operator amenable, a closed ideal of$A(G)$is weakly completely complemented in$A(G)$if and only if it has an approximate identity bounded in the$\cb$-multiplier norm. Keywords:$\cb$-multiplier norm, Fourier algebra, operator amenability, weak amenabilityCategories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25 38. CJM 2007 (vol 59 pp. 614) Labuschagne, C. C. A.  Preduals and Nuclear Operators Associated with Bounded,$p$-Convex,$p$-Concave and Positive$p$-Summing Operators We use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of$p$-convex,$p$-concave and positive$p$-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators. Keywords:$p$-convex operator,$p$-concave operator,$p$-summing operator, Banach space, Banach lattice, nuclear operator, sequence spaceCategories:46B28, 47B10, 46B42, 46B45 39. CJM 2007 (vol 59 pp. 638) MacDonald, Gordon W.  Distance from Idempotents to Nilpotents We give bounds on the distance from a non-zero idempotent to the set of nilpotents in the set of$n\times n$matrices in terms of the norm of the idempotent. We construct explicit idempotents and nilpotents which achieve these distances, and determine exact distances in some special cases. Keywords:operator, matrix, nilpotent, idempotent, projectionCategories:47A15, 47D03, 15A30 40. CJM 2007 (vol 59 pp. 393) Servat, E.  Le splitting pour l'opÃ©rateur de Klein--Gordon: une approche heuristique et numÃ©rique Dans cet article on \'etudie la diff\'erence entre les deux premi\eres valeurs propres, le splitting, d'un op\'erateur de Klein--Gordon semi-classique unidimensionnel, dans le cas d'un potentiel sym\'etrique pr\'esentant un double puits. Dans le cas d'une petite barri\ere de potentiel, B. Helffer et B. Parisse ont obtenu des r\'esultats analogues \a ceux existant pour l'op\'erateur de Schr\"odinger. Dans le cas d'une grande barri\ere de potentiel, on obtient ici des estimations des tranform\'ees de Fourier des fonctions propres qui conduisent \a une conjecture du splitting. Des calculs num\'eriques viennent appuyer cette conjecture. Categories:35P05, 34L16, 34E05, 47A10, 47A70 41. CJM 2006 (vol 58 pp. 859) Read, C. J.  Nonstandard Ideals from Nonstandard Dual Pairs for$L^1(\omega)$and$l^1(\omega)$The Banach convolution algebras$l^1(\omega)$and their continuous counterparts$L^1(\bR^+,\omega)$are much studied, because (when the submultiplicative weight function$\omega$is radical) they are pretty much the prototypic examples of commutative radical Banach algebras. In cases of nice'' weights$\omega$, the only closed ideals they have are the obvious, or standard'', ideals. But in the general case, a brilliant but very difficult paper of Marc Thomas shows that nonstandard ideals exist in$l^1(\omega)$. His proof was successfully exported to the continuous case$L^1(\bR^+,\omega)$by Dales and McClure, but remained difficult. In this paper we first present a small improvement: a new and easier proof of the existence of nonstandard ideals in$l^1(\omega)$and$L^1(\bR^+,\omega)$. The new proof is based on the idea of a nonstandard dual pair'' which we introduce. We are then able to make a much larger improvement: we find nonstandard ideals in$L^1(\bR^+,\omega)$containing functions whose supports extend all the way down to zero in$\bR^+$, thereby solving what has become a notorious problem in the area. Keywords:Banach algebra, radical, ideal, standard ideal, semigroupCategories:46J45, 46J20, 47A15 42. CJM 2006 (vol 58 pp. 548) Galanopoulos, P.; Papadimitrakis, M.  Hausdorff and Quasi-Hausdorff Matrices on Spaces of Analytic Functions We consider Hausdorff and quasi-Hausdorff matrices as operators on classical spaces of analytic functions such as the Hardy and the Bergman spaces, the Dirichlet space, the Bloch spaces and$\BMOA$. When the generating sequence of the matrix is the moment sequence of a measure$\mu$, we find the conditions on$\mu$which are equivalent to the boundedness of the matrix on the various spaces. Categories:47B38, 46E15, 40G05, 42A20 43. CJM 2005 (vol 57 pp. 1249) Lindström, Mikael; Saksman, Eero; Tylli, Hans-Olav  Strictly Singular and Cosingular Multiplications Let$L(X)$be the space of bounded linear operators on the Banach space$X$. We study the strict singularity andcosingularity of the two-sided multiplication operators$S \mapsto ASB$on$L(X)$, where$A,B \in L(X)$are fixed bounded operators and$X$is a classical Banach space. Let$1 Categories:47B47, 46B28

44. CJM 2005 (vol 57 pp. 771)

Schrohe, E.; Seiler, J.
 The Resolvent of Closed Extensions of Cone Differential Operators We study closed extensions $\underline A$ of an elliptic differential operator $A$ on a manifold with conical singularities, acting as an unbounded operator on a weighted $L_p$-space. Under suitable conditions we show that the resolvent $(\lambda-\underline A)^{-1}$ exists in a sector of the complex plane and decays like $1/|\lambda|$ as $|\lambda|\to\infty$. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of $\underline A$. As an application we treat the Laplace--Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem $\dot{u}-\Delta u=f$, $u(0)=0$. Keywords:Manifolds with conical singularities, resolvent, maximal regularityCategories:35J70, 47A10, 58J40

45. CJM 2005 (vol 57 pp. 506)

Gross, Leonard; Grothaus, Martin
 Reverse Hypercontractivity for Subharmonic Functions Contractivity and hypercontractivity properties of semigroups are now well understood when the generator, $A$, is a Dirichlet form operator. It has been shown that in some holomorphic function spaces the semigroup operators, $e^{-tA}$, can be bounded {\it below} from $L^p$ to $L^q$ when $p,q$ and $t$ are suitably related. We will show that such lower boundedness occurs also in spaces of subharmonic functions. Keywords:Reverse hypercontractivity, subharmonicCategories:58J35, 47D03, 47D07, 32Q99, 60J35

46. CJM 2005 (vol 57 pp. 225)

Booss-Bavnbek, Bernhelm; Lesch, Matthias; Phillips, John
 Unbounded Fredholm Operators and Spectral Flow We study the gap (= projection norm'' = graph distance'') topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary. Categories:58J30, 47A53, 19K56, 58J32

47. CJM 2005 (vol 57 pp. 61)

 On Operators with Spectral Square but without Resolvent Points Decompositions of spectral type are obtained for closed Hilbert space operators with empty resolvent set, but whose square has closure which is spectral. Krein space situations are also discussed. Keywords:unbounded operators, closed operators,, spectral resolution, indefinite metricCategories:47A05, 47A15, 47B40, 47B50, 46C20

48. CJM 2004 (vol 56 pp. 742)

Jiang, Chunlan
 Similarity Classification of Cowen-Douglas Operators Let $\cal H$ be a complex separable Hilbert space and ${\cal L}({\cal H})$ denote the collection of bounded linear operators on ${\cal H}$. An operator $A$ in ${\cal L}({\cal H})$ is said to be strongly irreducible, if ${\cal A}^{\prime}(T)$, the commutant of $A$, has no non-trivial idempotent. An operator $A$ in ${\cal L}({\cal H})$ is said to a Cowen-Douglas operator, if there exists $\Omega$, a connected open subset of $C$, and $n$, a positive integer, such that (a) ${\Omega}{\subset}{\sigma}(A)=\{z{\in}C; A-z {\text {not invertible}}\};$ (b) $\ran(A-z)={\cal H}$, for $z$ in $\Omega$; (c) $\bigvee_{z{\in}{\Omega}}$\ker$(A-z)={\cal H}$ and (d) $\dim \ker(A-z)=n$ for $z$ in $\Omega$. In the paper, we give a similarity classification of strongly irreducible Cowen-Douglas operators by using the $K_0$-group of the commutant algebra as an invariant. Categories:47A15, 47C15, 13E05, 13F05

49. CJM 2004 (vol 56 pp. 277)

Dostanić, Milutin R.
 Spectral Properties of the Commutator of Bergman's Projection and the Operator of Multiplication by an Analytic Function It is shown that the singular values of the operator $aP-Pa$, where $P$ is Bergman's projection over a bounded domain $\Omega$ and $a$ is a function analytic on $\bar{\Omega}$, detect the length of the boundary of $a(\Omega)$. Also we point out the relation of that operator and the spectral asymptotics of a Hankel operator with an anti-analytic symbol. Category:47B10

50. CJM 2004 (vol 56 pp. 134)

Li, Chi-Kwong; Sourour, Ahmed Ramzi
 Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States Every norm $\nu$ on $\mathbf{C}^n$ induces two norm numerical ranges on the algebra $M_n$ of all $n\times n$ complex matrices, the spatial numerical range $$W(A)= \{x^*Ay : x, y \in \mathbf{C}^n,\nu^D(x) = \nu(y) = x^*y = 1\},$$ where $\nu^D$ is the norm dual to $\nu$, and the algebra numerical range $$V(A) = \{ f(A) : f \in \mathcal{S} \},$$ where $\mathcal{S}$ is the set of states on the normed algebra $M_n$ under the operator norm induced by $\nu$. For a symmetric norm $\nu$, we identify all linear maps on $M_n$ that preserve either one of the two norm numerical ranges or the set of states or vector states. We also identify the numerical radius isometries, {\it i.e.}, linear maps that preserve the (one) numerical radius induced by either numerical range. In particular, it is shown that if $\nu$ is not the $\ell_1$, $\ell_2$, or $\ell_\infty$ norms, then the linear maps that preserve either numerical range or either set of states are `inner'', {\it i.e.}, of the form $A\mapsto Q^*AQ$, where $Q$ is a product of a diagonal unitary matrix and a permutation matrix and the numerical radius isometries are unimodular scalar multiples of such inner maps. For the $\ell_1$ and the $\ell_\infty$ norms, the results are quite different. Keywords:Numerical range, numerical radius, state, isometryCategories:15A60, 15A04, 47A12, 47A30
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