Restrictions of bounded linear operators: Closed range

Let be a bounded linear operator on a Banach space and let be a subspace of which is a Banach space and invariant. Denote by the restriction of to This paper explores the questions:

If the range of is closed, under what conditions is the range of closed?

If the range of is closed, under what conditions is the range of closed?

     

Laurent series for inversion of linearly perturbed bounded linear operators on Banach space

In this paper we find necessary and sufficient conditions for the existence of a Laurent series expansion with a finite order pole at the origin for the inverse of a linearly perturbed bounded linear operator mapping one Banach space to another. In particular we show that the inversion defines linear projections that separate the Banach spaces into corresponding complementary subspaces. We present two pertinent applications.     

Outer preserving linear operators

A natural question about linear operators on the Hilbert-Hardy space is answered, motivated by work in geophysical imaging. Namely, which bounded linear operators on the Hardy space preserve the set of all shifted outer functions? A complete characterization is determined, which allows an explicit construction of all such operators. Every operator that preserves the set of shifted outer functions is necessarily a product-composition operator, consisting of composition with a shifted outer function followed by multiplication with a (possibly different) shifted outer function. Such operators represent important physical processes, including the propagation of seismic wave energy through the earth. Applications to seismic imaging are briefly discussed.     

Invertibility of linear differential operators with closed range and poisson coefficients

The invertibility and injectivity properties of linear differential operators with closed range and Poisson coefficients are studied in the context of their equivalence in several spaces of vector functions defined on the axis. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 143–147, January, 1999.     

A hierarchy of generalized invariants for linear partial differential operators

We study invariants of linear partial differential operators in two variables under gauge transformations. Using the Beals-Kartashova factorization, we construct a hierarchy of generalized invariants for operators of an arbitrary order. We study the properties of these invariants and give some examples. We also show that the classic Laplace invariants correspond to some particular cases of generalized invariants. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 470–478, June, 2006.     

Necessary and sufficient conditions for the inversion of linearly-perturbed bounded linear operators on Banach space using Laurent series

Using a Laurent series representation, we present a detailed discussion of necessary and sufficient conditions for the inversion of linearly-perturbed bounded linear operators on Banach space that are singular in the unperturbed state.     

Norms of some operators on bounded symmetric domains

Let D be a bounded symmetric domain, H(D) the class of all holomorphic functions on D and u∈H(D). Operator norm of the multiplication operator on the weighted Bergman space , as well as of weighted composition operator from to a weighted-type space are calculated.     

A remark on numerical range of semi-hyponormal operators

Let U be the unilateral shift on ?². For any complex numbers α and β, put T = αU + βU* and S = T ². Then we show that the operator S is convexoid.     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.     

Bounded composition operators with closed range on the Dirichlet space

For composition operators on spaces of analytic functions it is well known that norm estimates can be converted to Carleson measure estimates. The boundedness of the composition operator becomes equivalent to a Carleson measure inequality. The measure corresponding to a composition operator on the Dirichet space is , where is the cardinality of the preimage . The composition operator will have closed range if and only if the corresponding measure satisfies a ``reverse Carleson measure' theorem: for all . Assuming is bounded, a necessary condition for this inequality is a reverse of the Carleson condition: (C) for all Carleson squares . It has long been known that this is not sufficient for a completely general measure. Here we show that it is also not sufficient for the special measures . That is, we construct a function such that is bounded and satisfies (C) but the composition operator does not have closed range.

     

Systems of linear ordinary differential equations with bounded coefficients may have very oscillating solutions

An elementary example shows that the number of zeroes of a component of a solution of a system of linear ordinary differential equations cannot be estimated through the norm of coefficients of the system.

     

On the Origin and Numerical Range of Bounded Operators

Numerical range has an important applications on spectrum distribution of operators. In this paper, we devoted to characterizing operators whose numerical range contains the origin. Some necessary and sufficient conditions are given by operator decomposition technique and constructive methods. Furthermore, the closeness of the numerical range of a given operator is also investigated.     

Polynomials generated by linear operators

We study the class of Banach algebra-valued -homogeneous polynomials generated by the powers of linear operators. We compare it with the finite type polynomials. We introduce a topology on similar to the weak topology, to clarify the features of these polynomials.

     

Norms of multiplication operators on Hardy spaces and weighted composition operators from Hardy spaces to weighted-type spaces on bounded symmetric domains

Let D be a bounded symmetric domain. We calculate operator norm of the multiplication operator on the Hardy space H^p(D), as well as of the weighted composition operator from H^p(D) to a weighted-type space.     

A new approach to factorization of a class of almost-periodic triangular symbols and related Riemann-Hilbert problems

The factorization of almost-periodic triangular symbols, G, associated to finite-interval convolution operators is studied for two classes of operators whose Fourier symbols are almost periodic polynomials with spectrum in the group αZ+βZ+Z (α,β∈]0,1[, α+β>1, α/β∉Q). The factorization problem is solved by a method that is based on the calculation of one solution of the Riemann-Hilbert problem GΦ₊=Φ₋ in L_∞(R) and does not require solving the associated corona problems since a second linearly independent solution is obtained by means of an appropriate transformation on the space of solutions to the Riemann-Hilbert problem. Some unexpected, but interesting, results are obtained concerning the Fourier spectrum of the solutions of GΦ₊=Φ₋. In particular it is shown that a solution exists with Fourier spectrum in the additive group αZ+βZ whether this group contains Z or not. Possible application of the method to more general classes of symbols is considered in the last section of the paper.     

Uniform factorization for compact sets of operators

We prove a factorization result for relatively compact subsets of compact operators using the Bartle and Graves Selection Theorem, a characterization of relatively compact subsets of tensor products due to Grothendieck, and results of Figiel and Johnson on factorization of compact operators. A further proof, essentially based on the Banach-Dieudonné Theorem, is included. Our methods enable us to give an easier proof of a result of W.H. Graves and W.M. Ruess.

     

Strong uniqueness of best approximations in spaces of bounded linear operators

The present paper is concerned with problems of the strong uniqueness of thebest approximation and the characterization of a uniqueness element in operator spaces.Some results on the strong uniqueness of the best approximation operator from RS-setsare proved and the uniqueness element of a sun in the compact operator space from c_0to c_0 is characterized by the strict Kolmogorov's condition. Some recent results due toLewicki and others are extended and improved.     

Hua operators and Poisson transform for bounded symmetric domains

Let Ω be a bounded symmetric domain of non-tube type in Cn with rank r and S its Shilov boundary. We consider the Poisson transform Psf(z) for a hyperfunction f on S defined by the Poisson kernel Ps(z,u)=s(h(z,z)^n/r/²|h(z,u)^n/r|), (z,u)×Ω×S, s∈C. For all s satisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When Ω is the type I matrix domain in Mn,m(C) (n?m), we prove that an eigenvalue equation for the second order Mn,n-valued Hua operator characterizes the image.