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# Topic: Bounded linear operator

###### In the News (Sat 25 Oct 14)

 Bounded operator -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-07) Let us note that a bounded linear operator is not necessarily a (Click link for more info and facts about bounded function) bounded function, the latter would require that the norm of L(v) is bounded for all v. Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed (A rectangular array of elements (or entries) set out by rows and columns) matrix. A common procedure for defining a bounded linear operator between two given Banach spaces is as follows. www.absoluteastronomy.com /encyclopedia/b/bo/bounded_operator.htm   (517 words)

 [No title]   (Site not responding. Last check: 2007-11-07) The term bounded appears in different parts of mathematics where a notion of "size" can be given. A set S in a metric space (M, d) is bounded if it is contained in a ball of finite radius, i.e. A set S in a topological vector space is bounded if it is contained in some multiple of every basic neighbourhood of zero. www.online-encyclopedia.info /encyclopedia/b/bo/bounded.html   (227 words)

 MC243 Aspects of Linear Analysis To be able to investigate whether or not a linear operator between normed vector spaces is bounded, and determine its norm. Linear operator, continuous linear operator, bounded linear operator, equivalence of continuity and boundedness for linear operators, operator norm, unbounded linear operator, B(X, Y) is a normed vector space with the operator norm, examples to investigate whether or not a linear operator is bounded and determine its norm, proof for a bounded linear operator ,norm of a composition of bounded linear operators, concept of equivalent norms, to know that all norms on a finite-dimensional vector space are equivalent, example of inequivalent norms. www.mcs.le.ac.uk /Modules/Year3/MC243.html   (667 words)

 [No title] adjoint +------------------------------------------------------------ The adjoint of a bounded linear operator A on a Hilbert space is the unique operator B which satisfies (Ax,y)=(x,By) for all x,y in H. One calls the adjoint A^*. linear operator +------------------------------------------------------------ A linear operator is a linear map between two Hilbert spaces or two Banach spaces. norm +------------------------------------------------------------ The norm of a bounded linear operator A on a Hilbert space H is defined as A www.math.harvard.edu /~knill/sofia/data/functionalanalysis.txt   (355 words)

 Essentially Normal   (Site not responding. Last check: 2007-11-07) Let E be an infinite dimensional subspace of C(S), the space of bounded continuous functions on a localy compact Hausdorff space S. For a regular Borel measure m on S, each element of E can be regarded as a bounded linear operator on L^p(mu) for 1<=p The main result of this paper states that the strong operator topology thus induced on E is properly weaker than the strict topology. For E the space of bounded analytic functions on a plane region, and m=Lebesgue area measure on this region, this answers a question raised by Rubel and Shields in: The space of bounded analytic functions on a plane region (Ann. www.mth.msu.edu /~shapiro/Pubvit/Downloads/Noncoinc/Noncoinc.html   (136 words)

 Math 519 - Operator Algebras   (Site not responding. Last check: 2007-11-07) This course is an introduction to the theory of operator algebras. Operator theory: spectral theorem for normal bounded operators. Math 516 (Linear Analysis) or equivalent and some familiarity with algebraic concepts such as algebras, ideals, etc., are sufficient as prerequisites for the course. www.math.ualberta.ca /~runde/math519.html   (197 words)

 Joan Cerd&gravea;: selected papers We consider interpolation of operators acting on functions that belong to a given cone $Q$ with the so--called decomposition property. In previous papers, the authors have established a unified method for the study of the commutators of bounded linear operators and certain operators in interpolation theory. Rochbert and Weiss developed the study of commutators of bounded linear operators and certain operators, generally unbounded and nonlinear obtained by complex interpolation. www.mat.ub.es /~cerda/JCselec.html   (1588 words)

 MA5524 HW #11   (Site not responding. Last check: 2007-11-07) Construct two border examples for the Open Mapping Theorem (4.12-2), one which shows that completeness of X and Y is necessary and one which shows that surjectivity of the linear operator is necessary (i.e., T must be onto''). Give an example of a bounded linear operator from a normed space X onto a normed space Y which is not an open mapping. Give an example of a bounded linear operator from a Banach space X into a Banach space Y which is not an open mapping. www.math.mtu.edu /~trolson/hw11.html   (104 words)

 Nathan S. Feldman's Home Page Definition: An operator T is countably hypercyclic if there is a bounded separated sequence with dense orbit. In particular, one may easily use adjoints of multiplication operators on Bergman or Hardy spaces to construct countably hypercyclic operators that are not hypercyclic. The hypercyclicity criterion basically says if an operator has two dense sets Y and Z (we may assume they are actually linear subspaces) such that the forward orbits go to zero on Y and on Z backward orbits go to zero, then our operator is hypercyclic. home.wlu.edu /~feldmann/Papers/CountablyHC.html   (361 words)

 Department of Mathematics, University of Strathclyde Banach space adjoint of a bounded linear operator and its relation with the Hilbert space adjoint. Algebra of compact linear operators, finite rank operators are compact, norm limit of compact operators is compact. Applications to linear algebraic equations and integral equations of the second kind. www.maths.strath.ac.uk /ungrad/classes/922.htm   (321 words)

 Bounded 2-linear operators on 2-normed sets   (Site not responding. Last check: 2007-11-07) In this paper properties of bounded 2-linear operators from a 2-normed set into a normed space are considered. The space of these operators is a Banach space and a symmetric 2-normed space. In the third part we will formulate Banach-Steinhaus Theorems for a family of bounded 2-linear operators from a 2-normed set into a Banach space. www.math.hr /glasnik/vol_39/no2_11.html   (73 words)

 Norms of Operators from Lp into Lq in the Real and the Complex Case ) is also bounded and we are interested in the exact relation between the norm of the operator T in the real case and in the complex case. In analysis the considered relation has, for instance, its applications in the interpolation theory and connection with the constants from the fundamental Grothendieck inequality. In addition, we show that the norms of the operators acting between spaces of functions of bounded p-variation and the norms of the positive linear operators between L epubl.luth.se /1402-1617/2004/170   (264 words)

 Functional analysis quiz   (Site not responding. Last check: 2007-11-07) Consider the strong, uniform, and weak, topologies on the space of all bounded linear operators on an infinite-dimensional Hilbert space. The spectrum of an operator is the whole closed disk of radius equal to the uniform operator of the operator As usual, let R(z) be the resolvent, meaning the inverse of T-z for T an operator and for z not in the spectrum of T. www.math.umn.edu /~garrett/m/fun/q/8q/q.shtml   (181 words)

 Department of Mathematics   (Site not responding. Last check: 2007-11-07) \item Define the adjoint operator $T^*$ of a given bounded linear operator and deduce from the previous result that $T^*$ is a bounded linear operator. You are also reminded that if $U$ is a unitary operator, i.e.\ $UU^*=U*U=I$ then its spectrum $\sigma(U)$ is a subset of the unit circle $\{\lambda; \lambda=1\}$.] \end{question} \begin{question} Show that any bounded linear operator $T$ may be expressed uniquesly in the form $T=A+iB$ where $A$ and $B$ are Hermitian operators. Define a \underline{normal} operator and establish a condition on $A$ and $B$ in the above form for $T$ to be normal. www.york.ac.uk /depts/maths/exams/fnanal/exam76.htm   (789 words)

 Comprehensive Exams   (Site not responding. Last check: 2007-11-07) A linear operator is bounded iff it is continuous Finite dimension implies all linear operators are bounded Eigenvectors of a self adjoint operator associated with distinct eigenvalues are orthogonal www.math.uri.edu /~quinn/comps.html   (72 words)

 Exponential Stability of an Abstract Nondissipative Linear System on a Hilbert space ${\cal H}$, where A is skew-adjoint, B is bounded, and $\varepsilon$ is a positive parameter. We also obtain a Hautus-type criterion for exact controllability of system (A, G), where G is a bounded linear operator from another Hilbert space to ${\cal H}$. Our result about the stability is then applied to establish the exponential stability of several elastic systems with indefinite viscous damping, as well as the exponential stabilization of the elastic systems with noncolocated observation and control. epubs.siam.org /sam-bin/dbq/article/36493   (216 words)

 No Title is a bounded linear operator on a Banach space X, the operator e (In quantum mechanics, we are interested in operators such that [A,B] is a multiple of the identity. Unfortunately, it can be easily shown that such a pair of operators cannot both be bounded, so the proof of BCH breaks down.) www.uwm.edu /~kevinm/texfiles/bch/bch.html   (254 words)

 The generalized condition numbers of bounded linear operators in Banach spaces   (Site not responding. Last check: 2007-11-07) The generalized condition numbers of bounded linear operators in Banach spaces These condition numbers may be applied to the perturbation analysis for the solution of ill-posed differential equations and bounded linear operator equations in infinite dimensional Banach spaces. Different expressions for the two generalized condition numbers are discussed in this paper and applied to the perturbation analysis of the operator equation. www.austms.org.au /Publ/Jamsa/V76P2/n59.html   (89 words)

 Numerical Ranges and Dilations (ResearchIndex)   (Site not responding. Last check: 2007-11-07) We also investigate the possibilities of dilating an operator A to operators with simple structure under the assumption that W (A) is included in a special region. 5 Structure of operators with numerical radius one (context) - Ando - 1973 1 the numerical range of a bounded operator (context) - Donoghue, On - 1957 citeseer.ist.psu.edu /38173.html   (262 words)

 An Operator Bound Related To Regular Operators (ResearchIndex)   (Site not responding. Last check: 2007-11-07) Regular operators on L p -spaces are characterised in terms of an operator bound which is associated with certain generalisations of the Feynman-Kac formula. L 2 (\Sigma; S; ¯; C) be a bounded linear operator. Let Q be the spectral measure acting on L 2 (\Sigma; S; ¯; C) of multiplication by the characteristic functions of elements of the oe-algebra S, so that if f is a bounded S-measurable function, Q(f) =... citeseer.ist.psu.edu /41843.html   (245 words)

 Nathan S. Feldman's Home Page   (Site not responding. Last check: 2007-11-07) Theorem 1: Suppose that T is a bounded linear operator on a Banach space X. If there is a d > 0 and a vector whose orbit comes within a distance d of every point in X, then T is hypercyclic. Theorem 1 allows us to give a more general form of the well known Hypercyclicity Criterion. Theorem 3: If T is a bounded linear operator on a Banach space X and x is a vector with dense orbit under T and y is a vector satisfying that the closure of Orb(T,y) is countable and compact, then (x+y) also has dense orbit under T. home.wlu.edu /~feldmann/Papers/PerturbHCVectors.html   (273 words)

 Atlas: On copies of c_0 in the bounded linear operator space by Jose M. Amigo   (Site not responding. Last check: 2007-11-07) The particular case when the function space considered is L(X, Y), the Banach space of bounded linear operators provided with the operator norm, will be addressed in this talk. These results can be extended to the space of all bounded vector measures. The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caey-82. atlas-conferences.com /cgi-bin/abstract/caey-82   (281 words)

 Schur Products of Operators and the Essential Numerical Range   (Site not responding. Last check: 2007-11-07) Abstract: Given an orthonormal basis E for a separable infinite-dimensional Hilbert space H, the Schur product of two bounded linear operators A and B on H with respect to E is the operator whose matrix entries are obtained by taking the termwise product of the matrix entries for A and B. It can be shown that the Schur product is a bounded linear operator on H, and hence Schur multiplication defines a new, commutative Banach algebra B There is a basis such that Schur multiplication by T is a compact operator in B(B(H)) mapping Schatten classes into smaller Schatten classes. www.eecs.umich.edu /~qstout/abs/TAMS81.html   (197 words)

 Proceedings of the American Mathematical Society Abstract: In a Banach space, Gelfand's formula is used to find the spectral radius of a continuous linear operator. In this paper, we show another way to find the spectral radius of a bounded linear operator in a complete topological linear space. We also show that Gelfand's formula holds in a more general setting if we generalize the definition of the norm for a bounded linear operator. www.ams.org /proc/1998-126-01/S0002-9939-98-04430-X/home.html   (228 words)

 On Properties of the Powers of a Bounded Linear Operator and their Characterization by its Spectrum an Resolvent ...   (Site not responding. Last check: 2007-11-07) On Properties of the Powers of a Bounded Linear Operator and their Characterization by its Spectrum an Resolvent Helmuth Ch. Titel On Properties of the Powers of a Bounded Linear Operator and their Characterization by its Spectrum an Resolvent Helmuth Ch. On On Properties Properties of of the the Powers Powers of of a a Bounded Bounded Linear Linear Operator Operator and and their their Characterization Characterization by by its its Spectrum Spectrum an an Resolvent Resolvent Helmuth Helmuth Ch Ch R%C3%83%C2%B6nnefarth R%C3%83%C2%B6nnefarth Tectum Tectum Verlag Verlag eis.clathrat.de /On_Properties_of_the_Powers_of_a_Bounded_Linear_Operator_and_their_Characterization_by_its_Spectrum_an_Resolvent_Helmuth_Ch._Rönnefarth_Tectum_Verlag.html   (149 words)

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