Operators In Lp Research Articles

The Commutant of the Laplace Operator in Lp(R)

AbstractLet L = ‐ d2/dx2 be the Laplace operator in Lp(R), 1<p < ∞. It is shown that a bounded operator T in Lp(R) commutes with L, in the sense that the domain D of L is T‐invariant and TLf = LTf for each f in D, if and only if T commutes with all (bounded) multiplier operators corresponding to symmetric p‐multipliers on R. The bicommutant L consists of all the symmetric p‐multiplier operators.

Measures on spaces of operators and isometries

One considers a question that arises at the investigation of isometric operators in vector-valued Lp -spaces. Let E, F be Banach spaces, let p>0, let μ be a probability Borel measure on the space on continuous linear operators from E into F such that for any e e E one has $$\left\| e \right\|^p = \int {\left\| {Te} \right\|^p } d\mu \left( T \right)$$ . In the cases when: 1) E=C(K), K is a metric compactum, F is an arbitrary space, p>1 and 2) 2)E=F=Lq,p>1, q>1 q ∉[p,2] it is Proved that the support of the measure μ is contained in the set of the operators that are scalar multiples of isometries. For E=C(K) one obtains an isomorphic analogue of this result: if the Banach-Mazur distance between C(K) and the P -sum of Banach spaces is small, then the distance between C(K) and one of the spaces is small.

Invariant subspaces of weighted shift operators

Let s be a weighted shift operator in lp, p e[1+∞)∶s(b0, b1, ...)=(0, λ0, b0, λ1, b1, ...). One proves its unicellularity under the condition ¦λi¦ ↓ 0 and also under some weaker conditions. One obtains also unicellularity conditions for weighted shift operators in Banach spaces cf numerical sequences. One gives a new proof of the following theorem of M. P. Thomas: if (Π i=0 n−1 ¦λi¦)1/n /ad 0 and ¦λi¦=0(i−e), ɛ > 0, then the operator s is unicellular in ip. One considers also a multiple weighted shift, corresponding to the case when bi are finite-dimensional vectors. Under the condition μi+1∥b∥ ⩽ ∥ ⩽ μi∥b∥, μ ↓ 0 one obtains the description of the invariant subspaces of this operator, using formal matrix power series.

Isometric operators in vector-valued LP-spaces

The note is devoted to the generalization of A. I. Plotkin's theorem on the equimeasurability of a function and of its image under a LP-isometry to the case of isometric operators, acting in certain vector-valued LP-spaces. One obtains the description of the isometric operators in LP(X; C(K)).

The integral representation of linear operators in Lp

The integral representation of linear operators in Lp

Spectral projections of L1 operators in type III? von Neumann algebras

The Lp spaces of a type IIIλ factor are represented in terms of the discrete decomposition of the factor. The Lp operators are characterized through the notion of wandering projections, and those projections that are spectral projections of positive Lp operators are completely characterized. Several special cases are considered, including a description of those operators that are bounded parts of a positive Lp operator.

Boundedness of a class of vector-valued multiplier operators in Lp (T?)

Boundedness of a class of vector-valued multiplier operators in Lp (T?)

The Finite Section Method for Two-dimensional Wiener-Hopf Integral Operators in Lp with Piecewise Continuous Symbols

Mathematische NachrichtenVolume 116, Issue 1 p. 61-73 Article The Finite Section Method for Two-dimensional Wiener-Hopf Integral Operators in Lp with Piecewise Continuous Symbols Albrecht Böttcher, Albrecht Böttcher Technische Hochschule Karl-Marx-Stadt Sektion Mathematik DDR - 9010 Karl-Marx-Stadt PSF 964Search for more papers by this author Albrecht Böttcher, Albrecht Böttcher Technische Hochschule Karl-Marx-Stadt Sektion Mathematik DDR - 9010 Karl-Marx-Stadt PSF 964Search for more papers by this author First published: 1984 https://doi.org/10.1002/mana.19841160106Citations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume116, Issue11984Pages 61-73 RelatedInformation

The application of dissipative operators to nonlinear diffusion equations

with U(L, ~1) positive, smooth for z > 0 and other technical conditions. The initial value problem with o(u, u,) = a(u), o(O) = 0 and g continuous has been previously studied by a different approach; see [8]. In Section 2 we prove that, under the appropriate conditions, A, the closure of Au = (a(~, u,)u,)~, may be regarded as a dissipative operator in Lp which generates a contraction semigroup there. Burnell McKissick proved this for the case a(u, u,) = u(u) in L’ (private communication). Conditions are given, in Section 3, for the existence of a positive lower bound of u(& X) = e”g. g E D(A), in sets of f > 0, --co < x < co. In such a region it is shown, in Section 4, that u(t. x) is a classical solution. The advance over (81 is that, for u(u) independent of u,, u(O) = co is allowed if

Regular compact factorization of integral operators in Lp

Regular compact factorization of integral operators in Lp

Boundedness of the convolution operator in Lp(Zm) and smoothness of the symbol of the operator

Sufficient conditions for the boundedness of the convolution operator in Lp(Zm) are found. These conditions are imposed on the symbol of the operator in terms of the spaces Hα and Hβ (functions of bounded variation of order β). The results obtained here generalize the results of S. B. Stechkin and I. I. Hirschman [Ref. Zh. Mat.7, No. 7821 (I960)] for the one-dimensional case.

Intertwining Certain Operators in lp

Intertwining Certain Operators in <i>l^p </i>

Domains of fractional powers of ordinary differential operators in spaces Lp

An exact description (in terms of smoothness and specified boundary conditions) is obtained for the domains of fractional powers of an ordinary differential operator of arbitrary order with general boundary conditions, acting in spaces Lp.

On operators in Lp (Rn) spaces which commute with shifts

On operators in Lp (Rn) spaces which commute with shifts

Many-dimensional operators of convolution type in spaces of weight-integrable functions

We show that a convolution operator in the weight space Lp is similar to a generalized convolution operator in Lp. We obtain necessary and sufficient conditions for an operator of convolution type, acting in a weight space, to have the Noether property in a cone. These conditions say, in effect, that the operator symbol must not degenerate on the hull of some tubular domain associated with the weight and the cone.

On the relation between amenability of locally compact groups and the norms of convolution operators

l~p(#) f (x) = # * f (x) = .I f (Y-1 x) d#(y) . G The norm of zrp(#) as operator in LP(G) is denoted }l#H~p, and it is < 1. It is known that for a fixed p, 1 < p < oo, the group G is amenable if and only if II#ll. = 1 for all probability measures on G (cf. Greenleaf [3], p. 48). Furthermore it was shown by Day [1], that a countable discrete group is amenable, if there exists a probability measure # whose support contains e and generates G and satisfies I[#ll, = 1. The aim of the present paper is to generalize Day's result to arbitrary locally compact groups, that is to prove the following result:

Uniformly bounded projections and semi-groups of operators

Classes of operators called uniformly bounded projections defined on Banach spaces have been formally introduced (see for example (2), 8·9·28; 8·9·92), and their connexions with semi-groups of operators considered. Notable cases of such operators include the Dirichlet integrals which are connected with the semi-group of Poisson operators in Lp In the main conclusions of this paper, there is an exposition of the connexion between classes of uniformly bounded projections and semi-groups of operators. In particular, it is shown that certain uniformly bounded analytic semi-groups of operators give rise to classes of uniformly bounded projections. Various of the known special cases in Lp are also considered as Fourier trans form multiplier operators with characteristic functions of intervals as multipliers.

Invariant subspaces, similarity and isometric equivalence of certain commuting operators inLp

Invariant subspaces, similarity and isometric equivalence of certain commuting operators in<i>L</i>_<i>p</i>

On semigroups generated by restrictions of elliptic operators to invariant subspaces

Let A be the closed unbounded operator inLp(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ1≦argλ≦θ2} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1<p<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2≦θ<θ2≦3π/2 ; and there exists a constantc such that (I)││(λ-A)-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE− andE+ such thatA is completely reduced by the direct sum decompositionLp(G)=E−Lp (G) ⊕E+Lp (G) and each of the operatorsAE− and—AE+ is the infinitestimal generator of an analytic semigroup.

Commutators, 𝐶^{𝑘}-classification, and similarity of operators

We generalize the results of our recent paper, The C k {C^k} -classification of certain operators in L p {L_p} . II, to the abstract setting of a pair of operators satisfying the commutation relation [ M , N ] = N 2 [M,N] = {N^2} .