scholarly journals Finite rank intermediate Hankel operators and the big Hankel operator

2006 ◽  
Vol 2006 ◽  
pp. 1-3 ◽  
Author(s):  
Tomoko Osawa
Keyword(s):  
Bergman Space ◽  
Finite Rank ◽  
Hankel Operator ◽  
Closed Subspace ◽  
Hankel Operators ◽  
Coordinate Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

LetLa2be a Bergman space. We are interested in an intermediate Hankel operatorHφMfromLa2to a closed subspaceMofL2which is invariant under the multiplication by the coordinate functionz. It is well known that there do not exist any nonzero finite rank big Hankel operators, but we are studying same types in caseHφMis close to big Hankel operator. As a result, we give a necessary and sufficient condition aboutMthat there does not exist a finite rankHφMexceptHφM=0.

scholarly journals Toeplitz Operators on the Bergman Space of Planar Domains with Essentially Radial Symbols

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Roberto C. Raimondo
Keyword(s):  
Bergman Space ◽  
Toeplitz Operators ◽  
Boundary Behavior ◽  
Berezin Transform ◽  
Planar Domain ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Planar Domains ◽  
Necessary And Sufficient ◽  
The Relationship

We study the problem of the boundedness and compactness of when and is a planar domain. We find a necessary and sufficient condition while imposing a condition that generalizes the notion of radial symbol on the disk. We also analyze the relationship between the boundary behavior of the Berezin transform and the compactness of

scholarly journals THE FINITE SUM OF THE PRODUCTS OF TWO TOEPLITZ OPERATORS

2009 ◽  
Vol 86 (1) ◽  
pp. 45-60 ◽  
Author(s):  
XUANHAO DING
Keyword(s):  
Toeplitz Operator ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Rank Perturbation ◽  
Sum Of Products ◽  
Finite Sum ◽  
Necessary And Sufficient

AbstractWe consider in this paper the question of when the finite sum of products of two Toeplitz operators is a finite-rank perturbation of a single Toeplitz operator on the Hardy space over the unit disk. A necessary condition is found. As a consequence we obtain a necessary and sufficient condition for the product of three Toeplitz operators to be a finite-rank perturbation of a single Toeplitz operator.

scholarly journals ON THE COMPLEXITY OF FINDING A NECESSARY AND SUFFICIENT CONDITION FOR BLASCHKE-OSCILLATORY EQUATIONS

2014 ◽  
Vol 57 (3) ◽  
pp. 543-554
Author(s):  
JANNE HEITTOKANGAS ◽  
ATTE REIJONEN
Keyword(s):  
Differential Equation ◽  
Bergman Space ◽  
Nontrivial Solution ◽  
Unit Disc ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Number Of Zeros ◽  
Zero Sequence ◽  
Zeros Of Solutions ◽  
Necessary And Sufficient

AbstractIf A(z) belongs to the Bergman space , then the differential equation f″+A(z)f=0 is Blaschke-oscillatory, meaning that the zero sequence of every nontrivial solution satisfies the Blaschke condition. Conversely, if A(z) is analytic in the unit disc such that the differential equation is Blaschke-oscillatory, then A(z) almost belongs to . It is demonstrated that certain “nice” Blaschke sequences can be zero sequences of solutions in both cases when A ∈ or A ∉ . In addition, no condition regarding only the number of zeros of solutions is sufficient to guarantee that A ∈ .

scholarly journals Finite-rank intermediate Hankel operators on the Bergman space

2001 ◽  
Vol 25 (1) ◽  
pp. 19-31 ◽  
Author(s):  
Takahiko Nakazi ◽  
Tomoko Osawa
Keyword(s):  
Bergman Space ◽  
Orthogonal Projection ◽  
Finite Rank ◽  
Lebesgue Space ◽  
Unit Disc ◽  
Hankel Operator ◽  
Hankel Operators ◽  
Weighted Bergman Space ◽  
Open Unit ◽  
Open Unit Disc

LetL2=L2(D,r dr dθ/π)be the Lebesgue space on the open unit disc and letLa2=L2∩ℋol(D)be the Bergman space. LetPbe the orthogonal projection ofL2ontoLa2and letQbe the orthogonal projection ontoL¯a,02={g∈L2;g¯∈La2,   g(0)=0}. ThenI−P≥Q. The big Hankel operator and the small Hankel operator onLa2are defined as: forϕinL∞,Hϕbig(f)=(I−P)(ϕf)andHϕsmall(f)=Q(ϕf)(f∈La2). In this paper, the finite-rank intermediate Hankel operators betweenHϕbigandHϕsmallare studied. We are working on the more general space, that is, the weighted Bergman space.

scholarly journals Multipliers of invariant subspaces in the bidisc

1994 ◽  
Vol 37 (2) ◽  
pp. 193-199 ◽  
Author(s):  
Takahiko Nakazi
Keyword(s):  
Invariant Subspace ◽  
Hankel Operator ◽  
Invariant Subspaces ◽  
Inner Function ◽  
Finite Codimension ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inner Functions ◽  
Necessary And Sufficient

For any nonzero invariant subspace M in H2(T2), set . Then Mx is also an invariant subspace of H2(T2) that contains M. If M is of finite codimension in H2(T2) then Mx = H2(T2) and if M = qH2(T2) for some inner function q then Mx = M. In this paper invariant subspaces with Mx = M are studied. If M = q1H2(T2) ∩ q2H2(T2) and q1, q2 are inner functions then Mx = M. However in general this invariant subspace may not be of the form: qH2(T2) for some inner function q. Put (M) = {ø ∈ L ∞: ø M ⊆ H2(T2)}; then (M) is described and (M) = (Mx) is shown. This is the set of all multipliers of M in the title. A necessary and sufficient condition for (M) = H∞(T2) is given. It is noted that the kernel of a Hankel operator is an invariant subspace M with Mx = M. The argument applies to the polydisc case.

Composition operators between Bergman spaces of logarithmic weights

2015 ◽  
Vol 26 (09) ◽  
pp. 1550068 ◽  
Author(s):  
Ern Gun Kwon ◽  
Jinkee Lee
Keyword(s):  
Integral Representation ◽  
Bergman Space ◽  
Composition Operators ◽  
Counting Function ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Logarithmic Weight

Let [Formula: see text] be the composition operator induced by a holomorphic self-map φ of the open complex unit disk. In this paper, a necessary and sufficient condition for the boundedness of [Formula: see text] from one weighted Bergman space of logarithmic weight into another is described in terms of a growth condition of a generalized counting function for φ. We make use of a new integral representation of a modified counting function which depends on log-convexity of the weight function as well as some estimates for the norm of the weighted Bergman space.

scholarly journals On the extension of positive linear functionals

2000 ◽  
Vol 23 (1) ◽  
pp. 31-35 ◽  
Author(s):  
Ergashboy Muhamadiev ◽  
Adel T. Diab
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Closed Subspace ◽  
Sufficient Condition ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Necessary And Sufficient Condition ◽  
Semi Group ◽  
Positive Linear Functionals ◽  
Necessary And Sufficient

A necessary and sufficient condition to extend a continuous linear real functionals which is positive with respect to a semi-group defined on a subspace of a linear space is discussed in this paper. The case of a closed subspace of a Banach space is also discussed.

scholarly journals Some Properties of Interior and Closure in General Topology

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 624
Author(s):  
Soon-Mo Jung ◽  
Doyun Nam
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Set ◽  
Open Set ◽  
Necessary And Sufficient ◽  
Open Subspace

We present the necessary and sufficient conditions that the intersection of an open set and a closed set becomes either an open set or a closed set. As their dualities, we further introduce the necessary and sufficient conditions that the union of a closed set and an open set becomes either a closed set or an open set. Moreover, we give some necessary and sufficient conditions for the validity of U ∘ ∪ V ∘ = ( U ∪ V ) ∘ and U ¯ ∩ V ¯ = U ∩ V ¯ . Finally, we introduce a necessary and sufficient condition for an open subset of a closed subspace of a topological space to be open. As its duality, we also give a necessary and sufficient condition for a closed subset of an open subspace to be closed.

scholarly journals The moment spaces of normed linear spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 277-283
Author(s):  
Fowzi Ahmad Sejeeni ◽  
Matooq Ahmad Badri
Keyword(s):  
Normed Linear Space ◽  
Closed Subspace ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Independent Sequence ◽  
Linear Spaces ◽  
Linearly Independent ◽  
The Moment ◽  
Moment Spaces ◽  
Necessary And Sufficient

For a linearly independent sequence in a normed linear space the moment space is defined. Basic properties of moment spaces are discussed as well as a necessary and sufficient condition for the moment space to be a closed subspace of l∞.

CONSTRUCTION OF DUAL g-FRAMES FOR CLOSED SUBSPACES

2011 ◽  
Vol 09 (06) ◽  
pp. 947-964 ◽  
Author(s):  
BAI-YUN YU ◽  
ZHI-BIAO SHU
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Closed Subspace ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

In this paper, we introduce dual g-frames for a closed subspace in a separable Hilbert space and also give a characterization. Generally dual g-frames for a closed subspace are non-commutative. Therefore, we construct dual g-frames for closed subspaces from two aspects and give the corresponding formulas, respectively. Finally, we give a necessary and sufficient condition for commutative dual g-frame pairs for closed subspaces under certain conditions.

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