Beurling's theorem and invariant subspaces for the shift on Hardy spaces

Let G be a bounded open subset in the complex plane and let H~2(G) denote the Hardy space on G. We call a bounded simply connected domain W perfectly connected if the boundary value function of the inverse of the Riemann map from W onto the unit disk D is almost 1-1 with respect to the Lebesgue measure on D and if the Riemann map belongs to the weak-star closure of the polynomials in H~∞(W). Our main theorem states: in order that for each M∈Lat (M_z), there exist u∈H~∞(G) such that M=∨{uH~2(G)}, it is necessary and sufficient that the following hold: (1) each component of G is a perfectly connected domain; (2) the harmonic measures of the components of G are mutually singular; (3) the set of polynomials is weak-star dense in H~∞(G). Moreover, if G satisfies these conditions, then every M∈Lat (M_z) is of the form uH~2(G), where u∈H~∞(G) and the restriction of u to each of the components of G is either an inner function or zero.     

Orthonormal bases and quasi-splitting subspaces in pre-Hilbert spaces

Let S be a pre-Hilbert space. We study quasi-splitting subspaces of S and compare the class of such subspaces, denoted by Eq(S), with that of splitting subspaces E(S). In [D. Buhagiar, E. Chetcuti, Quasi splitting subspaces in a pre-Hilbert space, Math. Nachr. 280 (5-6) (2007) 479-484] it is proved that if S has a non-zero finite codimension in its completion, then Eq(S)≠E(S). In the present paper it is shown that if S has a total orthonormal system, then Eq(S)=E(S) implies completeness of S. In view of this result, it is natural to study the problem of the existence of a total orthonormal system in a pre-Hilbert space. In particular, it is proved that if every algebraic complement of S in its completion is separable, then S has a total orthonormal system.     

Characteristic functions and joint invariant subspaces

Let T:=[T₁,…,Tn] be an n-tuple of operators on a Hilbert space such that T is a completely non-coisometric row contraction. We establish the existence of a “one-to-one” correspondence between the joint invariant subspaces under T₁,…,Tn, and the regular factorizations of the characteristic function ΘT associated with T. In particular, we prove that there is a non-trivial joint invariant subspace under the operators T₁,…,Tn, if and only if there is a non-trivial regular factorization of ΘT. We also provide a functional model for the joint invariant subspaces in terms of the regular factorizations of the characteristic function, and prove the existence of joint invariant subspaces for certain classes of n-tuples of operators.We obtain criteria for joint similarity of n-tuples of operators to Cuntz row isometries. In particular, we prove that a completely non-coisometric row contraction T is jointly similar to a Cuntz row isometry if and only if the characteristic function of T is an invertible multi-analytic operator.     

Rank-one commutators on invariant subspaces of the Hardy space on the bidisk

We discuss on invariant subspaces of H²(Γ²) on which .     

Free pluriharmonic majorants and commutant lifting

In this paper we initiate the study of sub-pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball

     

Entire functions, analytic continuation, and the fractional parts of a linear function

The main result of the paper is as follows.Theorem. Suppose that G(z) is an entire function satisfying the following conditions: 1) the Taylor coefficients of the function G(z) are nonnegative: 2) for some fixed C>0 and A>0 and for |z|>R₀, the following inequality holds:

Blaschke products and the rank of backward shift invariant subspaces over the bidisk

Let H²(D²) be the Hardy space over the bidisk. For sequences of Blaschke products {φn(z):−∞<n<∞} and {ψn(w):−∞<n<∞} satisfying some additional conditions, we may define a Rudin type invariant subspace M. We shall determine the rank of H²(D²)?M for the pair of operators and .     

On complete-cocomplete subspaces of an inner product space

In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space S is complete if and only if there exists a -additive state on C(S), the orthomodular poset of complete-cocomplete subspaces of S. We then consider the problem of whether every state on E(S), the class of splitting subspaces of S, can be extended to a Hilbertian state on E( ); we show that for the dense hyperplane S (of a separable Hilbert space) constructed by P. Pták and H. Weber in Proc. Am. Math. Soc. 129 (2001), 2111–2117, every state on E(S) is a restriction of a state on E( ).This revised version was published online in April 2005 with a corrected missing date string.     

Zero-based subspaces and quasi-invariant subspaces of the Bargmann-Fock space

We use the Weierstrass σ-function associated with a lattice in the complex plane to construct finite dimensional zero-based subspaces and quasi-invariant subspaces of given index in the Bargmann-Fock space.     

An extremal problem for even positive definite entire functions of exponential type

We consider an extremal problem for even positive definite entire functions of exponential type with zero mean with power weight on the semiaxis. This problem is related to the multidimensional Jackson-Stechkin theorem in the space L ₂(?ⁿ).     

Some weighted estimates for Littlewood-Paley functions and radial multipliers

We prove some weighted estimates for certain Littlewood-Paley operators on the weighted Hardy spaces H_w^p (0<p?1) and on the weighted L^p spaces. We also prove some weighted estimates for the Bochner-Riesz operators and the spherical means.     

On subspaces of C[0, 1] consisting of nonsmooth functions

For each Hölder spa?e H ^ω, we construct an infinite-dimensional closed subspace G of C[0, 1], isomorphic to l ¹ and such that, for each function x ∈ G not identically zero, its restriction to the set of positive measure does not belong to the Hölder space H ^ω.     

<Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-estimates of Littlewood-Paley and Lusin functions for Jacobi analysis

For α≥β≥ -1/2 let Δ(x) = (2shx)2α+1(2chx)2β+1 denote the weight function on R+ and L1(Δ) the space of integrable functions on R+ with respect to Δ(x)dx, equipped with a convolution structure. For a suitable φ∈ L1(Δ), we put φt(x) = t-1Δ(x)-1Δ(x/t)φ(x/t) for t > 0 and define the radial maximal operator Mφ as usual manner. We introduce a real Hardy space H1(Δ) as the set of all locally integrable functions f on R+ whose radial maximal function Mφ(f) belongs to L1(Δ). In this paper we obtain a relation between...     

Construction of an atomic decomposition for functions with compact support

Chang, Krantz and Stein [D.-C. Chang, S.G. Krantz, E.M. Stein, Hp theory on a smooth domain in Rn and elliptic boundary value problems, J. Funct. Anal. 114 (1993) 286-347] proved that if f∈Hp(Rn) and f vanishes outside , then f has an atomic decomposition whose atoms are contained in Ω. The purpose of this paper is to give another proof for the case n/(n+1)<p?1 and Ω a cube. Our argument provides a simple, direct construction of the desired atomic decomposition, and it works in a class of function spaces more general than the usual Hardy spaces.     

Polynomiality criterion for entire functions of several complex variables

The “radial” polynomiality criterion for entire functions of several complex variables is proved. Translated fromMatermaticheskie Zametki, Vol. 66, No. 4, pp. 500–502, October, 1999.     

Normal Families and Uniqueness of Entire Functions and Their Derivatives

Let f be a nonconstant entire function; let k ≥ 2 be a positive integer; and let a be a nonzero complex number. If f（z） = a→f′（z） = a, and f′（z） = a →f^（k）（z） = a, then either f = Ce^λz ＋ a or f = Ce^λz ＋ a（λ - 1）/）λ, where C and ）, are nonzero constants with λ^k-1 = 1. The proof is based on the Wiman-Vlairon theory and the theory of normal families in an essential way.