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In this paper, we characterize that the boundedness, compactness and spectral structure of dual Toeplitz operators acting on the orthogonal complement of the harmonic Bergman space. This generalizes the corresponding results for dual Toeplitz operators on the orthogonal complement of the Bergman space due to Stroethoff and Zheng’s paper [Trans. Amer. Math. Soc., 354, 2495–2520 (2002)].

Let H2(ⅅ2) be the Hardy space over the bidisk ⅅ2, and let Mψ,ϕ = [(ψ(z) − ϕ(w))2] be the submodule generated by (ψ(z) − ϕ(w))2, where ψ(z) and ϕ(w) are nonconstant inner functions. The related quotient module is denoted by Nψ,ϕ = H2(ⅅ2) ⊖ Mψ,ϕ. In this paper, we give a complete characterization for the essential normality of Nψ,ϕ. In particular, if ψ(z)= z, we simply write Mψ,ϕ and Nψ,ϕ as Mϕ and Nϕ respectively. This paper also studies compactness of evaluation operators L(0)∣nϕ and R(0)ϕnϕ, essential spectrum of compression operator Sz on Nϕ, essential normality of compression operators Sz and Sw on Nϕ.

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number $\sqrt{q}$. This establishes a key step that one may extend our previous explicit calculations of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of genera and ideal complexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the results of Yu on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigation., Comment: 23 pages. Section 5.4 corrected

Zhao and Ho asked in a recent paper that for each T0 space X, whether KB(X) (the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Topκ of the category Top0 of T0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Topκ.

In 2002, in the paper entitled “A subspace theorem approach to integral points on curves”, Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt’s subspace theorem in Diophantine approximation. Since then, the program has led a great progress in the study of Diophantine approximation. It is known that the counterpart of Schmidt’s subspace in Nevanlinna theory is H. Cartan’s Second Main Theorem. In recent years, the method of Corvaja and Zannier has been adapted by a number of authors and a big progress has been made in extending the Second Main Theorem to holomorphic mappings from C into arbitrary projective variety intersecting general divisors by using H. Cartan’s original theorem. We call such method “a Cartan’s Second Main Theorem approach”. In this survey paper, we give a systematic study of such approach, as well as survey some recent important results in this direction including the recent work of the author with Paul Voja.

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli–Chern class on compact complex manifolds, and proved that the (1, 1) curvature form of the Levi–Civita connection represents the first Aeppli–Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi–Civita Ricci-flat metrics and classify minimal complex surfaces with Levi–Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi–Civita Ricci-flat metrics are Kahler Calabi–Yau surfaces and Hopf surfaces.

In my former paper “A pre-order principle and set-valued Ekeland variational principle” (see [J. Math. Anal. Appl., 419, 904–937 (2014)]), we established a general pre-order principle. From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles (denoted by EVPs) in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy’s EVP in [On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381–396 (2011)], where the perturbation contains a weak τ-function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle, but also can be applied to obtain the above Khanh and Quy’s EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.

In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the Lp boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space Llog L(Sn−1). Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.

The purpose of this paper is to study a new viscosity iterative algorithm based on a generalized contraction for finding a common element of the set of solutions of a general variational inequality problem for finite inversely strongly accretive mappings and the set of common fixed points for a countable family of strict pseudo-contractions in uniformly smooth Banach spaces. We prove some strong convergence theorems under some suitable conditions. The results obtained in this paper improve and extend the recent ones announced by many others in the literature.

In this paper, we discuss the Schatten-p class (0 < p ≤ ∞) of Toeplitz operators on generalized Fock space with the symbol in positive Borel measure. It makes a great difference from other papers by using the estimates of the kernel and the weight together instead of separately estimating each other. We also get the equivalent conditions when a Toeplitz operator is in the Schatten-p class.

Let X, Y be Banach spaces and M be a linear subspace in X × Y = {{x, y}|x ∈ X, y ∈ Y}. We may view M as a multi-valued linear operator from X to Y by taking M(x) = {y|{x, y} ∈ M}. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M. The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces.

Our first purpose in this paper is to provide necessary conditions for a weak*-closed translation invariant subspace in the semigroup algebra of a locally compact topological foundation semigroup to be completely complemented. We give conditions when a weak*-closed left translation invariant subspace in Ma(S)* of a compact cancellative foundation semigroup S is the range of a weak*-weak* continuous projection on Ma(S)* commuting with translations. Let G be a locally compact group and A be a Banach G-module. Our second purpose in this paper is to study some projections on A* and B(A*) which commutes with translations and convolution.

The purpose of this paper is by using CSQ method to study the strong convergence problem of iterative sequences for a pair of strictly asymptotically pseudocontractive mappings to approximate a common fixed point in a Hilbert space. Under suitable conditions some strong convergence theorems are proved. The results presented in the paper are new which extend and improve some recent results of Acedo and Xu [Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal., 67(7), 2258–2271 (2007)], Kim and Xu [Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal., 64, 1140–1152 (2006)], Martinez-Yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal., 64, 2400–2411 (2006)], Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279, 372–379 (2003)], Marino and Xu [Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl., 329(1), 336–346 (2007)], Osilike et al. [Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps. J. Math. Anal. Appl., 326, 1334–1345 (2007)], Liu [Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal., 26(11), 1835–1842 (1996)], Osilike et al. [Fixed points of demi-contractive mappings in arbitrary Banach spaces. Panamer Math. J., 12 (2), 77–88 (2002)], Gu [The new composite implicit iteration process with errors for common fixed points of a finite family of strictly pseudocontractive mappings. J. Math. Anal. Appl., 329, 766–776 (2007)].

The paper investigates Lp convergence and Marcinkiewicz-Zygmund strong laws of large numbers for random elements in a Banach space under the condition that the Banach space is of Rademacher type p, 1 < p < 2. The paper also discusses Lr convergence and Lr bound for random elements without any geometric restriction condition on the Banach space.

In this paper, we introduce the concept of a g-Besselian frame in a Hilbert space and discuss the relations between a g-Besselian frame and a Besselian frame. We also give some characterizations of g-Besselian frames. In the end of this paper, we discuss the stability of g-Besselian frames. Our results show that the relations and the characterizations between a g-Besselian frame and a Besselian frame are different from the corresponding results of g-frames and frames.

In this paper, we study the limit cycles bifurcations of four fine focuses in Z 4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By computing the Liapunov constants and periodic constants carefully, we show that for a certain Z 4-equivariant quintic systems, there are four fine focuses of five order and five limit cycles can bifurcate from each, we also find conditions of center and isochronous center for this system. The process of proof is algebraic and symbolic by using common computer algebra soft such as Mathematica, the expressions after being simplified in this paper are simple relatively. Moreover, what is worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is good for Z 4-equivariant quintic system and the results where multiple singular points become isochronous centers at the same time are less in published references.

In this paper, we prove the generalized Hyers-Ulam stability of homomorphisms in quasi-Banach algebras associated with the following Pexiderized Jensen functional equation % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGaemOzay2aaeWaaeaadaWcaaqaaiabdIha4fXafv3y % SLgzGmvETj2BSbacgaGae83kaSIaemyEaKhabaGae8Nmaidaaiab-T % caRiabdQha6bGaayjkaiaawMcaaiab-jHiTiabdEgaNnaabmaabaWa % aSaaaeaacqWG4baEcqWFsislcqWG5bqEaeaacqWFYaGmaaGae83kaS % IaemOEaOhacaGLOaGaayzkaaGae8xpa0JaemiAaGMae8hkaGIaemyE % aKNae8xkaKIae8Nla4caaa!5ABC! $$ f\left( {\frac{{x + y}} {2} + z} \right) - g\left( {\frac{{x - y}} {2} + z} \right) = h(y). $$ This is applied to investigating homomorphisms between quasi-Banach algebras. The concept of the generalized Hyers-Ulam stability originated from Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297–300 (1978).

In this paper, we prove the Hyers-Ulam-Rassias stability of isometric homomorphisms in proper CQ*-algebras for the following Cauchy-Jensen additive mapping: $$ 2f\left( {\frac{{x_1 + x_2 }} {2} + y} \right) = f(x_1 ) + f(x_2 ) + 2f(y). $$ The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias’ stability theorem that appeared in the paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300.

In this paper, we compute the derivations of the positive part of the two-parameter quantum group of type G2 by embedding it into a quantum torus. We also show that the first Hochschild cohomology group of this algebra is a two-dimensional vector space over the complex field.

By using the positive linear functional and the monotone subhomogeneous functional, including the general means and Riccati technique, some new oscillation criteria are established for the second order linear matrix differential system $$ (P(t)X'(t))' + R(t)X'(t) + Q(t)X(t) = 0, t \geqslant t_0 > 0, $$ where P(t), R(t), Q(t) are n × n real continuous matrix functions, P(t) and R(t) are commutative. The results improve and generalize those given in some previous papers, which can be seen by the examples given at the end of this paper.

The purpose of this paper is to study the convergence problem of the iteration scheme xn+1 = λn+1y +(1–λn+1) Tn+1xn for a family of infinitely many nonexpansive mappings T1, T2, . . . in a Hilbert space. It is proved that under suitable conditions this iteration scheme converges strongly to the nearest common fixed point of this family of nonexpansive mappings. The results presented in this paper extend and improve some recent results.

In this paper, we first prove that the θ-deformation U θ (2) of U(2) constructed by Connes and Violette is our special case of the quantum group U q (2) constructed in our previous paper. Then we will show that the set of traces on the C*–algebra U θ , θ irrational, is determined by the set of the traces on a subalgebra of U θ .

In this paper, we give a number of characterizations for a Banach space X which is isometric to a subspace of c0, or, c0(Γ), successively, in terms of extreme points of its dual unit ball BX*, Frechet and Gâteaux derivatives of its norm, or, in terms of w*-strongly exposed points and w*-exposed points of BX*.

The separability and the entanglement (that is, inseparability) of the composite quantum states play important roles in quantum information theory. Mathematically, a quantum state is a trace-class positive operator with trace one acting on a complex separable Hilbert space. In this paper, in more general frame, the notion of separability for quantum states is generalized to bounded positive operators acting on tensor product of Hilbert spaces. However, not like the quantum state case, there are different kinds of separability for positive operators with different operator topologies. Four types of such separability are discussed; several criteria such as the finite rank entanglement witness criterion, the positive elementary operator criterion and PPT criterion to detect the separability of the positive operators are established; some methods to construct separable positive operators by operator matrices are provided. These may also make us to understand the separability and entanglement of quantum states better, and may be applied to find new separable quantum states.

In this article, we classify 1-connected 8-dimensional Poincare complexes, topological manifolds and smooth manifolds whose integral homology groups are isomorphic to those of S3 × S5. A topic related to a paper of Escher and Ziller is also discussed.

Assume that m ≥, p is a prime number, (m, p(p – 1)) = 1, −1 ∉ 〈p〉 ⊂ (ℤ/mℤ)* and [(ℤ/mℤ)* : 〈p〉] = 4. In this paper, we calculate the value of Gauss sum \( G{\left( \chi \right)} = {\sum {_{{x \in \mathbb{F}^{ * }_{q} }} \chi {\left( x \right)}\zeta ^{{T{\left( x \right)}}}_{p} } } \) over \({\mathbb F}\) q , where q = p f , \( f = \frac{{\varphi {\left( m \right)}}} {4} \), χ is a multiplicative character of \({\mathbb F}\) q and T is the trace map from \({\mathbb F}\) q to \({\mathbb F}\) p . Under our assumptions, G(χ) belongs to the decomposition field K of p in ℚ(ζ m ) and K is an imaginary quartic abelian number field. When the Galois group Gal(K/ℚ) is cyclic, we have studied this cyclic case in another paper: "Gauss sums of index four: (1) cyclic case" (accepted by Acta Mathematica Sinica, 2003). In this paper we deal with the non–cyclic case.

The aim of the present paper is to investigate the existence of solutions to functional differential inclusions with infinite delay in Banach spaces. A relevant set of phase space axioms is proposed. The main tools used in this paper are certain fixed point theorems based on the setcontraction theory.

Let p be a prime, m ≥ 2, and (m, p(p – 1)) = 1. In this paper, we will calculate explicitly the Gauss sum $$ G{\left( \chi \right)} = {\sum {_{{x \in \mathbb{F}^{ * }_{q} }} \chi {\left( x \right)}\zeta ^{{T{\left( x \right)}}}_{p} } } $$ in the case of [(ℤ/mℤ)* : 〈p〉] = 4, and −1 ∉ 〈p〉, where q = p f , $$ f = \frac{{\varphi {\left( m \right)}}} {4} $$ , χ is a multiplicative character of $${\Bbb F}$$ q with order m, and T is the trace map for $${\Bbb F}$$ q / $${\Bbb F}$$ p . Under the assumptions [(ℤ/mℤ)* : 〈p〉] = 4 and −1 ∉ 〈p〉, the decomposition field of p in the cyclotomic field ℚ(ζ m ) is an imaginary quartic (abelian) field. And G (χ) is an integer in K. We deal with the case where K is cyclic in this paper and leave the non–cyclic case to the next paper.

This paper proves that, under the hypothesis g(t, 0, 0)≡0 and some natural assumptions, the generator g of a backward stochastic differential equation can be uniquely determined by the corresponding g-expectations with all terminal conditions. The main result of this paper also confirms and extends Peng Shige’s conjecture.

In this paper, we study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M. We first show that $$ \lim _{{r \to \infty }} \frac{{r^{2} }} {{V{\left( r \right)}}}{\int_{B{\left( r \right)}} {\Delta hdv{\kern 1pt} = {\kern 1pt} {\kern 1pt} 0} } $$ if h is a bounded subharmonic function. If we further assume that the Laplacian decays pointwisely faster than quadratically we show that h approaches its supremun pointwisely at infinity, under certain auxiliary conditions on the volume growth of M. In particular, our result applies to the case when the Riemannian manifold has maximum volume growth. We also derive a representation formula in our paper, from which one can easily derive Yau’s Liouville theorem on bounded harmonic functions.

In this paper, some new generalizations of inverse type Hilbert-Pachpatte integral inequalities are proved. The results of this paper reduce to those of Pachpatte (1998, J. Math. Anal. Appl. 226, 166–179) and Zhao and Debnath (2001, J. Math. Anal. Appl. 262, 411–418).

Denote a semisimple Banach algebra with an identity e by A. This paper studies the Fredholm, Weyl and Browder spectral theories in a semisimple Banach algebra, and meanwhile considers the properties of the Fredholm element, the Weyl element and the Browder element. Further, for a ∈ A, we give the Weyl’s theorem and the Browder’s theorem for a, and characterize necessary and sufficient conditions that both a and f(a) satisfy the Weyl’s theorem or the Browder’s theorem, where f is a complex-valued function analytic on a neighborhood of σ(a). In addition, the perturbations of the Weyl’s theorem and the Browder’s theorem are investigated.

The well-known Hartman-Grobman Theorem says that a C1 hyperbolic diffeomorphism F can be locally linearized by a homeomorphism Φ. For parameterized systems Fθ, known results show that the corresponding homeomorphisms Φθ exist uniquely in a functional space equipped with the supremum norm and depend continuously on the parameter θ. In this paper, we further extend the results to Holder dependence of Φθ on θ by Pugh’s strategy, but introducing a kind of special Holder norm instead of the usual supremum norm in the proof to control the linear parts of Fθ. This requires a new Holder linearization result for every Fθ.

Recently, Geher and Semrl have characterized the general form of surjective isometries of the set of all projections on an infinite-dimensional separable Hilbert space using unitaries and antiunitaries. In this paper, we study the surjective L2-isometries of the projection lattice of an infinite dimensional Hilbert space and show that every such isometry can also be described by unitaries and antiunitaries.

This paper is mainly about holomorphic mappings associated with conic regions which are closely connected with k − ST(α). We introduce new subclasses of starlike (spirallike) functions, namely, S (k,α)(S (k,α,β)), and discuss their coefficient estimates and the Fekete-Szego-Goluzin’s problem. Then we generalize S (k,α,β) on the unit ball Bn in ℂn, that is, k-conic spirallike mappings of type β and order α. We obtain the growth, covering and distortion theorems of the generalized mappings. Besides that, we construct k-conic spirallike mappings of type β and order α on Bn through Sc(k,α,β) by the generalized Roper-Suffridge extension operators.

Let G be a classical group over an arbitrary field F, acting on an n-dimensional vector space V = V(n, F) over a field F. In this paper, we classify the maximal subgroups of G, which normalizes a solvable subgroup N of GL(n, F) not lying in F*1V.

In this paper we introduce the notions of (Banach) density-equicontinuity and density-sensitivity. On the equicontinuity side, it is shown that a topological dynamical system is density-equicontinuous if and only if it is Banach density-equicontinuous. On the sensitivity side, we introduce the notion of density-sensitive tuple to characterize the multi-variant version of density-sensitivity. We further look into the relation of sequence entropy tuple and density-sensitive tuple both in measure-theoretical and topological setting, and it turns out that every sequence entropy tuple for some ergodic measure on an invertible dynamical system is density-sensitive for this measure; and every topological sequence entropy tuple in a dynamical system having an ergodic measure with full support is density-sensitive for this measure.

This paper is mainly devoted to proving the four equivalent defining properties of a CBA(κ) space. The proof is based on an interesting tool we established which describes the cyclical five-step deformation procedure for quadrangles in the model 2-plane $$S_\kappa^2$$ , including the limit shape of each step. As a byproduct we give a complete list of cyclical deformation procedures for all kinds of quadrangles on $$S_1^2$$ . At last we make a contrast of geometric properties of CBA with CBB spaces, including a comparison between their defining properties and a discussion about Alexandrov’s Lemma.

In this paper, we show that every super weakly compact convex subset of a Banach space is an absolute uniform retract, and that it also admits the uniform compact approximation property. These can be regarded as extensions of Lindenstrauss and Kalton’s corresponding results.

In this paper we prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points when the global sign is -1, using a recent work of the author on one divisibility of Iwasawa-Greenberg main conjecture for Rankin-Selberg p-adic L-functions. As a byproduct we also prove the equality for the above mentioned main conjecture under some local conditions., Comment: 13 Pages. Comments welcome

In this paper, we prove that for every symplectic matrix M possessing eigenvalues on the unit circle, there exists a symplectic matrix P such that P −1 MP is a symplectic matrix of the normal forms defined in this paper.

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial ℤ-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial ℤ-groups, we finally obtain the structure and invariants of the group G.

In this paper, we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation $${{\cal S}^b}: = {\overline {\rm{\Delta }} ^{\alpha /2}} + b \cdot \nabla $$ where $${\overline {\rm{\Delta }} ^{\alpha /2}}$$ is the truncated fractional Laplacian, α ∈ (1, 2) and b ∈ K −1 . In the second part, for a more general finite range jump process, we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance ∣x − y∣ in short time.

In this paper, let m ≥ 1 be an integer, M be an m-dimensional compact Riemannian manifold. Firstly the linearized Poincare map of the Lagrangian system at critical point x $${d \over {dt}}{L_q}\left( {t,x,\dot x} \right) - {L_p}\left( {t,x,\dot x} \right) = 0$$ is explicitly given, then we prove that Morse index and Maslov-type index of x are well defined whether the manifold M is orientable or not via the parallel transport method which makes no appeal to unitary trivialization and establish the relation of Morse index and Maslov-type index, finally derive a criterion for the instability of critical point and orientation of M and obtain the formula for two Maslov-type indices.

In this paper, using the method of blow-up analysis, we obtained a Trudinger—Moser inequality involving Lp-norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger—Moser functional.

In this paper we extend the Fibonacci-like maps to a wider class with the so-called “bounded combinatorics”. The Fibonacci-like renormalization operator $${\cal R}$$ is defined and we show that the orbit of each map from this class converges to a universal limit under iterates of $${\cal R}$$ .

Let f and g be functions in Fock—Sobolev space F2,m. In this paper, we completely characterize the boundedness and compactness of $${H_{\mathop f\limits^ - }}{T_{\mathop g\limits^ - }}$$ .

Let (M,g) be a compact Kahler manifold and (E,F) be a holomorphic Finsler vector bundle of rank r ≥ 2over M. In this paper, we prove that there exists a Kahler metric Φ defined on the projective bundle P(E)of E, which comes naturally from g and F. Moreover, a necessary and sufficient condition for Φ having positive scalar curvature is obtained, and a sufficient condition for Φ having positive Ricci curvature is established.

In this paper, assuming there is a fiberwise Morse function, we extend Bismut-Lott index theorem to the complex valued case.

In this paper, we study complex symmetric C0-semigroups on the Bergman space A2(ℂ+) of the right half-plane ℂ+. In contrast to the classical case, we prove that the only involutive composition operator on A2(ℂ+) is the identity operator, and the class of J-symmetric composition operators does not coincide with the class of normal composition operators. In addition, we divide semigroups {ϕt} of linear fractional self-maps of ℂ+ into two classes. We show that the associated composition operator semigroup {Tt} is strongly continuous and identify its infinitesimal generator. As an application, we characterize Jσ-symmetric C0-semigroups of composition operators on A2(ℂ+).