Norms of linear-fractional composition operators

We obtain a representation for the norm of the composition operator on the Hardy space whenever is a linear-fractional mapping of the form . The representation shows that, for such mappings , the norm of always exceeds the essential norm of . Moreover, it shows that a formula obtained by Cowen for the norms of composition operators induced by mappings of the form has no natural generalization that would yield the norms of all linear-fractional composition operators. For rational numbers and , Cowen's formula yields an algebraic number as the norm; we show, e.g., that the norm of is a transcendental number. Our principal results are based on a process that allows us to associate with each non-compact linear-fractional composition operator , for which \Vert C_\phi\Vert _e$">, an equation whose maximum (real) solution is . Our work answers a number of questions in the literature; for example, we settle an issue raised by Cowen and MacCluer concerning co-hyponormality of a certain family of composition operators.

     

Components of linear-fractional composition operators

We determine when two linear-fractional composition operators on the Hardy space H² belong to the same component in the collection of all composition operators on H². We show that two such composition operators in the same component may fail to have compact difference, which answers a question raised by Joel Shapiro and Carl Sundberg.     

Quadratic operator inequalities and linear-fractional relations

We study properties of solution sets of inequalities of the form

$X^* AX + B^* X + X^* B + C \leqslant 0,$

, where A, B, and C are bounded Hilbert space operators and A and C are self-adjoint. The following properties are considered: closedness and inferior points in Standard operator topologies, convexity, and nonemptiness.

     

Which linear-fractional composition operators are essentially normal?

We characterize the essentially normal composition operators induced on the Hardy space H² by linear-fractional maps; they are either compact, normal, or (the nontrivial case) induced by parabolic nonautomorphisms. These parabolic maps induce the first known examples of nontrivially essentially normal composition operators. In addition, we characterize those linear-fractionally induced composition operators on H² that are essentially self-adjoint, and present a number of results for composition operators induced by maps that are not linear-fractional.     

Operator linear-fractional relations: main properties, some applications

We consider operator linear-fractional relations of the form $$ F(K)=\left\{ {Q:A+BK=Q\left( {C+DK} \right)} \right\}, $$ where A, B, C, D, K, and Q are operators between Hilbert spaces. If C + DK is invertible, the relation F becomes a linear-fractional transformation. In the case where F is the automorphism of a unit operator ball, we study the conditions for F to be represented in the form of a composition of an automorphism and an affine relation. The results obtained are applied to the Abel–Schröder equations, Königs embedding problem, and some other questions.     

On the convexity of images of nonlinear integral operators

We study continuous nonlinear Urysohn-type integral operators acting from the spaces of vector functions with integrable components to the space of continuous functions. We obtain conditions under which the images of sets defined by pointwise constraints have a convex closure under the action of these operators. The result is used to justify a method of constructive approximation of these images and to derive a necessary solvability condition for Urysohn-type integral equations. A numerical method for finding the residual of equations of this type on the sets under consideration is justified.     

Commutation relations and hypercyclic operators

In this paper we establish hypercyclicity of continuous linear operators on \({H(\mathbb{C})}\) that satisfy certain commutation relations.     

Families of operators satisfying relations

A problem of unitary classification of families of operators R_i= R _i ^* =R_i/–1 in a Hilbert space, connected by some additional relations. Such families occur in problems concerning representations of a side class of *-algebras, among others, two parameter deformations U(su (2)), constructed by E. K. Sklyannyi.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 309–317, March, 1992.     

Compositions of linear-fractional transformations

We study the asymptotic behavior of the compositions (S_n o...o S₁)(z) and (S₁ o...o S_n)(z) of linear-fractional transformations S_n (z) (n=1,2,...) whose fixed points have limits. In particular, if S _n (z)=α _n (β _n +z)^-1, then the sequency of compositions (S₁o...o S_n)(z) at the point z=0 coincides with the sequence of convergents of the formal continued fraction $$\frac{{\alpha _1 }}{{\beta _1 + \frac{{\alpha _2 }}{{\beta _2 + \cdot \cdot \cdot }}}}.$$ The result obtained can be applied in the study of convergence of formal continued fractions.     

Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space H 2

In 1999 Nina Zorboska and in 2003 P. S.Bourdon, D. Levi, S.K.Narayan and J.H. Shapiro investigated the essentially normal composition operator ${C_\varphi }$ , when φ is a linear-fractional self-map of D. In this paper first, we investigate the essential normality problem for the operator T _w ${C_\varphi }$ on the Hardy space H ², where w is a bounded measurable function on ?D which is continuous at each point of F(φ), φ ∈ S(2), and T _w is the Toeplitz operator with symbol w. Then we use these results and characterize the essentially normal finite linear combinations of certain linear-fractional composition operators on H ².     

Some properties of a linear-fractional transformation

Seven properties of a linear-fractional analytic function, many of which are also valid in the domain of real variables, are pointed out. In either case, these properties are important for applications to problems of subterranean hydromechanics.     

Unbounded self-adjoint operators connected by algebraic relations

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 12, pp.1664–1668, December, 1989.