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Topic: Bounded operator

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	Bounded operator - Wikipedia, the free encyclopedia
		In functional analysis (a branch of mathematics), a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X.
		Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix.
		First, define a linear operator on a dense subset of the domain, such that it is locally bounded.
	en.wikipedia.org /wiki/Bounded_operator (343 words)

	Compact operator - Wikipedia, the free encyclopedia
		In functional analysis, a compact operator (or completely continuous operator) is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y.
		Any L that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalisation of the class of finite rank operators in an infinite-dimensional setting.
		The origin of the theory of compact operators is in the theory of integral equations.
	en.wikipedia.org /wiki/Compact_operator (475 words)

	Boundedness - Wikipedia, the free encyclopedia
		The term bounded appears in different parts of mathematics where a notion of "size" can be given.
		A linear transformation is bounded if the image of the unit ball is a bounded set.
		A partially ordered set is bounded if it is has both a greatest element and a least element.
	www.wikipedia.org /wiki/Bounded (251 words)

	Encyclopedia: Compact operator (Site not responding. Last check: 2007-11-07)
		In mathematics, a Fredholm operator is a bounded linear operator between two Banach spaces whose range is closed and whose kernel and cokernel are finite-dimensional.
		In mathematics, in particular functional analysis, singular values, or s-numbers of an bounded operator T acting on a Hilbert space are defined as the eigenvalues of (T*T)1/2.
		In mathematics, a nuclear operator or a trace-class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis.
	www.nationmaster.com /encyclopedia/Compact-operator (1182 words)

	Self-adjoint operator - SmartyBrain Encyclopedia and Dictionary (Site not responding. Last check: 2007-11-07)
		For infinite dimensional Hilbert spaces H, the structure of self-adjoint operators is complicated somewhat by the fact that the operators may be partial functions, that is defined on a proper subset of H.
		By the Hellinger-Toeplitz theorem, a symmetric everywhere defined operators is bounded.
		The spectrum of any bounded symmetric operator operator is real; in particular all its eigenvalues are real, although a symmetric operator may not have any eigenvalues.
	smartybrain.com /index.php/Self-adjoint_operator (1530 words)

	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.
		Define the operator L:X→X which acts by taking the ((linguistics) a word that is derived from another word) derivative, so it maps a polynomial P to its derivative P′.
	www.absoluteastronomy.com /encyclopedia/b/bo/bounded_operator.htm (517 words)

	Operator norm
		Operator Norm is a norm defined over the linear operators space between two Banach spaces (i.e.
		Given linear operator where both X,Y are Banach spaces with norms we define
		In general the operator norm of a bounded linear transformation L from V to W, where V and W are both normed real vector spaces (or both normed complex) vector spaces is defined as the supremum of the
	pedia.newsfilter.co.uk /wikipedia/o/op/operator_norm.html (677 words)

	Operator norm -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, the operator norm is a means to measure the "size" of certain (An operator that obeys the distributive law: A(f+g) = Af + Ag (where f and g are function)) linear operators.
		Because of this property, the continuous linear operators are also known as (Click link for more info and facts about bounded operator) bounded operators.
		The operator norm is indeed a norm on the space of all (Click link for more info and facts about bounded operator) bounded operators between V and W.
	www.absoluteastronomy.com /encyclopedia/o/op/operator_norm.htm (764 words)

	PlanetMath: closed operator (Site not responding. Last check: 2007-11-07)
		A core of a closable operator is a subset
		Cross-references: converge, inverse, bounded linear operator, properties, subset, restriction, operator, graph, sequence, linear operator, Banach space
		This is version 5 of closed operator, born on 2003-07-28, modified 2005-09-11.
	planetmath.org /encyclopedia/ClosedOperator.html (135 words)

	COUNTEREXAMPLES IN OPERATOR THEORY (Site not responding. Last check: 2007-11-07)
		Operators of arbitrary large norms that are bounded by 1 on a given basis of a separable infinite dimensional Hilbert space H. OT12.dvi
		A sequence of quasi-nilpotent operators acting on a Hilbert space with a norm limit whose spectral radius is 1.
		A sequence of nilpotent operators on H which converges with respect to the norm topology on B(H) to an operator which is not topologically nilpotent.
	web.um.ac.ir /~moslehian/cfa/OT.HTM (454 words)

	PlanetMath: operator norm (Site not responding. Last check: 2007-11-07)
		must be the zero operator and is assigned zero norm.
		satisfies all the properties of a norm and hence is called the operator norm (or the induced norm) of
		This is version 8 of operator norm, born on 2002-06-03, modified 2005-05-23.
	planetmath.org /encyclopedia/InducedNorm.html (122 words)

	PlanetMath: bounded operator (Site not responding. Last check: 2007-11-07)
		Thus the operator norm is the smallest constant
		Cross-references: bounded, finite-dimensional, vector spaces, between, theorem, zero operator, identity operator, mappings, zero map, zero vector space, constant, operator norm, real number, linear map, normed vector spaces
		This is version 13 of bounded operator, born on 2003-10-15, modified 2004-11-02.
	planetmath.org /encyclopedia/BoundedOperator.html (111 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^*.
		Important examples are bounded linear operators, linear operators which also continuous maps.
		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)

	The Ultimate Weak operator topology - American History Information Guide and Reference
		In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space such that the functional sending an operator T to the complex number
		The linear functionals on the set of bounded operators on a Hilbert space which are continuous in the strong operator topology are precisely those which are continuous in the WOT.
		Because of this fact, the closure of a convex set of operators in the WOT is the same as the closure of that set in the SOT.
	www.historymania.com /american_history/Weak_operator_topology (169 words)

	Articles - Self-adjoint operator (Site not responding. Last check: 2007-11-07)
		The structure of self-adjoint operators on infinite dimensional Hilbert spaces is complicated somewhat by the fact that the operators may be partial functions, meaning that they are defined only on a proper subspace of the Hilbert space.
		By the Hellinger-Toeplitz theorem, a symmetric everywhere defined operator is bounded.
		The operator theoretic adjoint P* of P is a restriction of the distributional extension of the formal adjoint.
	lastring.com /articles/Self-adjoint_operator?... (2200 words)

	Re: Bounded type (was Re: Range type)
		There's a very serious decision to be made: it is whether the bounded template classes' semantics are only to hold the invariant of the value being within bounds, or whether the bounded classes are also strongly typed, with no implicit conversions.
		The choices for the method of specifying bounds of 'bounded': a) Pass pointers or references to objects, whose values determine the bounds, b) pass a class defining bounds as the template's actual parameter, c) pass function objects defining bounds as the tempalte's actual parameters, d) pass the bounds as the template class constructor's arguments.
		bounded could be defined as: template < typename s, typename b, typename v = decltype(s()()), /* ^^^^^^^^^^^^^^^^^^ / typename c = std::less > class bounded;* I didn't use the default type for v, because it's not clear whether the type should be deduced from s or b - bot are equally good.
	www.talkaboutprogramming.com /group/comp.lang.c++.moderated/messages/171029.html (766 words)

	FuncAna
		Remark: the kernel of a bounded operator is always closed while the range may be not closed.
		A bounded operator is an orthogonal projector if and only if it is idempotent and self-adjoint.
		Norm of a self-adjoint operator is equal to the supremum of the absolute value of its quadratic form on the unit sphere.
	www.math.ttu.edu /~vshubov/FuncAna/FuncAna.html (947 words)

	LMS Proceedings Abstract, paper PLMS 1539 (Site not responding. Last check: 2007-11-07)
		Uniform Abel--Kreiss boundedness and the extremal behaviour of the Volterra operator
		By means of sharp estimates of the $L^1$-norm of the $n$th partial sums of the generating function of the Laguerre polynomials on the unit circle, it is also proved that $I - V$ is uniformly Kreiss bounded on the spaces $L^p [0,1]$, for $1 \leq p \leq \infty$.
		It is also shown that, for general operators, uniform Abel boundedness characterizes Cesàro boundedness and, as a consequence, uniform Kreiss boundedness is characterized in terms of a Cesàro type boundedness of order 1.
	www.lms.ac.uk /publications/proceedings/abstracts/p1539a.html (278 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		operator const T_var () const; /// for in parameter.
		operator[] (CORBA::ULong slot) const; /** * The allocbuf function allocates a vector of T elements that can * be passed to the T *data constructor.
		/** * For bounded sequences, the maximum length is part of the type and * cannot be set or modified, while for unbounded sequences, the * default constructor also sets the maximum length to 0.
	siesta.cs.wustl.edu /~schmidt/ACE_wrappers/TAO/tao/Sequence_T.h (4872 words)

	Bounded Linear Operator
		If f is bounded it has a norm, denoted f, which is the lower bound of all the constants k that make f a bounded operator.
		Our linear operator is continuous on R; in fact it scales R by a fixed amount and embeds it in the range.
		Given a linear operator f on a normed vector space, f is continuous at a point, iff f is bounded, iff f is uniformly continuous everywhere.
	www.mathreference.com /top-ban,blo.html (616 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		When comparing one bounded to another we try // to work out the result at compile time if possible (and if the // compiler is smart enough).
		So if you had two sufficiently wide boundeds and // added them you could end up with a result that is negative (and // maybe outside the min..max range).
		The 'worst' operation you can do on a // bounded is multiplication, so you should be able to guarantee // no overflow if you check that // abs(X) ::const_max) // for X = min and for X = max.
	membled.com /work/todo/bounded/test.cpp (2048 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)

	The Biot-Savart operator and electrodynamics on bounded subdomains of the three-sphere.
		Specifically, the Biot-Savart operator applied to a "current" V is a right inverse to curl; thus BS is important in the study of curl eigenvalue energy-minimization problems in geometry and physics.
		The helicity of a vector field, a measure of the coiling of its flow, is expressed as an inner product of BS(V) with V. We find upper bounds for helicity on the three-sphere; our bounds are not sharp but we produce explicit examples within an order of magnitude.
		Applications of the Biot-Savart operator include plasma physics, geometric knot theory, solar physics, and DNA replication.
	repository.upenn.edu /dissertations/AAI3125886 (246 words)

	Completely Bounded Maps and Operator Algebras (Site not responding. Last check: 2007-11-07)
		In this book the reader is provided with a tour of the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, together with some of their main applications.
		The author assumes only that the reader has a basic background in functional analysis, and the presentation is self-contained and paced appropriately for graduate students new to the subject.
		Experts will also want this book for their library since the author illustrates the power of methods he has developed with new and simpler proofs of some of the major results in the area, many of which have not appeared earlier in the literature.
	assets.cambridge.org /052181/6696/description/0521816696_description.htm (141 words)

	Bounded Linear Operator is Dot Product (Site not responding. Last check: 2007-11-07)
		Since f is bounded, let k be a bound on f.
		The bound on f is no larger than u.
		The bound on such a function is u, and is realized when u is dotted with itself.
	www.mathreference.com /top-ban,lindot.html (222 words)

	Trace class (Site not responding. Last check: 2007-11-07)
		A bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases Ω of H; the sum $\sum_{x\in \Omega}$ is finite.
		A bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases Ω of H; the sum
		In this case, the sum is called the trace of A, denoted by tr(A) and is independent of the choice of the orthonormal bases.
	www.termsdefined.net /tr/trace-class.html (399 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)

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