Weak Cartan-type second main theorem for holomorphic curves

In this paper, a weak Cartan-type second theorem for holomorphic curve f ： C→P^n（C） intersecting hypersurfaces Dj, 1≤j≤q, in P^n（C） in general position with degree dj is given as follows： For every ε〉0, there exists a positive integer M such that ｜｜（q - （n ＋ 1） ε）Tf（r）≤∑j^q=1 1/dj Nf^M（r,Dj）＋o（Tf（r））, where ＂｜｜＂ means the estimate holds for all large r outside a set of finite Lebesgue measure.     

Difference analogues of the second main theorem for meromorphic functions in several complex variables

In this paper, we obtain difference analogues of the second main theorem for meromorphic functions in several complex variables from which difference analogues of Picard‐type theorems are also obtained. Our results are improvements or extensions of some recent results of papers [Proc. Royal Soc. Edinburgh Section A Math. 137 , 457–474 (2007); Comput. Meth. Funct. Theor. 12 , No. 1, 343–361 (2012)]. The method we used is very different from theirs.     

Cartan's conjecture with moving targets of same growth and effective Wirsing's theorem over function fields

We formulate a generalization of the Cartan's theorem and the Cartan's conjecture over function fields in the case where the coefficients of the linear forms can be nonconstant functions. This generalization is used to prove the Wirsing's theorem over function fields. Furthermore, the methods we use here provide effective results. Received September 4, 1998; in final form April 23, 1999 / Published online May 8, 2000     

A Coburn type theorem for Toeplitz operators on the Dirichlet space

A celebrated theorem of Coburn asserts that, on the setting of the Hardy space, if a Toeplitz operator is nonzero, then either it is one-to-one or its adjoint operator is one-to-one. In this paper, we show that an analogous result holds for Toeplitz operators acting on the Dirichlet space.     

Degeneracy of holomorphic curves in surfaces

Let X be a complex projective algebraic manifold of dimension 2 and let D1,…,Du be distinct irreducible divisors on X such that no three of them share a common point. Let f: C→X\(U1≤i≤uDi) be a holomorphic map. Assume that u≥4 and that there exist positive integers n1,…,nu, c such that ninj(Di.Dj) = c for all pairs i, j. Then f is algebraically degenerate, i.e. its image is contained in an algebraic curve on X.     

Lomonosov's invariant subspace theorem for multivalued linear operators

The famous Lomonosov's invariant subspace theorem states that if a continuous linear operator on an infinite-dimensional normed space ``commutes' with a compact operator i.e., then has a non-trivial closed invariant subspace. We generalize this theorem for multivalued linear operators. We also provide an application to single-valued linear operators.

     

On the Nochka-Chen-Ru-Wong proof of Cartan's conjecture

In 1982-1983, E. Nochka proved a conjecture of Cartan on defects of holomorphic curves in Pn relative to a possibly degenerate set of hyperplanes. This was further explained by W. Chen in his 1987 thesis, and subsequently simplified by M. Ru and P.-M. Wong in 1991. The proof involved assigning weights to the hyperplanes. This paper provides further simplification of the proof of the construction of the weights, by bringing back the use of the convex hull in working with the “Nochka diagram.”     

Algebroid functions, Wirsing's theorem and their relations

In this paper, we first point out a relationship between the Second Main Theorem for algebriod functions in Nevanlinna theory and Wirsing's theorem in Diophantine approximation. This motivates a unified proof for both theorems. The second part of this paper deals with “moving targets” problem for holomorphic maps to Riemann surfaces. Its counterpart in Diophantine approximation follows from a recent work of Thomas J. Tucker. In this paper, we point out Tucker's result in the special case of the approximation by rational points could be obtained by doing a “translation” and applying the corresponding result with fixed target. However, we could not completely recover Tucker's result concerning the approximation by algebraic points. In the last part of this paper, cases in higher dimensions are studied. Some partial results in higher dimensions are obtained and some conjectures are raised. Received August 26, 1997; in final form June 30, 1998     

On the Hahn-Banach theorem for groups

Let be the family of all groups such that under each subadditive functional there exists an additive functional. We show that the class is between the class of all amenable groups and the family of all groups for which the Hyers stability theorem for homomorphisms holds true. Next, we generalize the classical Hahn-Banach theorem to the class . Received: 6 May 2005     

A lifting theorem for unbounded quasinormal operators

Relationships between minimal normal extensions of spectral and cyclic type of an unbounded quasinormal operator are discussed and some properties such as, for example, tightness of such extensions are established. A Yoshino type criterion on the lifting of the strong commutant of an unbounded quasinormal operator is proved.     

An interpolation theorem for proper holomorphic embeddings

Given a Stein manifold x of dimension n > 1, a discrete sequence , and a discrete sequence where , there exists a proper holomorphic embedding satisfying f(a _j ) = b _j for every j = 1,2,... Forstnerič and Prezelj supported by grants P1-0291 and J1-6173, Republic of Slovenia. Kutzschebauch supported by Schweizerische National fonds grant 200021-107477/1. Ivarsson supported by The Wenner-Gren Foundations.     

Uniqueness theorem for algebraic curves on compact Riemann surfaces

In this paper, we prove a uniqueness theorem for algebraic curves from a compact Riemann surface into complex projective spaces.     

A local lifting theorem for subnormal operators

Criteria for the existence of lifts of operators intertwining subnormal operators are established. The main result of the paper reduces lifting questions for general subnormal operators to questions about lifts of cyclic subnormal operators. It is shown that in general the existence of local lifts (i.e. those coming from cyclic parts) for a pair of subnormal operators does not imply the existence of a global lift. However this is the case when minimal normal extensions of subnormal operators in question are star-cyclic.

     

On hypercyclic rank one perturbations of unitary operators

Recently, S. Grivaux showed that there exists a rank one perturbation of a unitary operator in a Hilbert space which is hypercyclic. Another construction was suggested later by the first and the third authors. Here, using a functional model for rank one perturbations of singular unitary operators, we give yet another construction of hypercyclic rank one perturbation of a unitary operator. In particular, we show that any countable union of perfect Carleson sets on the circle can be the spectrum of a perturbed (hypercyclic) operator.     

LITTLEWOOD-PALEY THEOREM FOR SCHROEDINGER OPERATORS

Let H be a Schroedinger operator on R^n. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.     

A restriction theorem for Métivier groups

In the spirit of an earlier result of D. Müller on the Heisenberg group we prove a restriction theorem on a certain class of two step nilpotent Lie groups. Our result extends that of Müller also in the framework of the Heisenberg group.     

On entire solutions of some type of nonlinear difference equations

In this article, the existence of finite order entire solutions of nonlinear difference equations f~n+ P_d(z, f) = p_1 e~(α1 z)+ p_2 e~(α2 z) are studied, where n ≥ 2 is an integer, Pd(z, f) is a difference polynomial in f of degree d(≤ n-2), p_1, p_2 are small meromorphic functions of ez, and α_1, α_2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.     

Toeplitz operators defined by sesquilinear forms: Fock space case

The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions.