Analogues of the “zero-two” law for positive linear contractions inL p andC(X)

LetT be a positive linear contraction inL ^p (1≦p<∞), then we show that lim ‖T ^pf −T ⁿ⁺¹ f‖ _p ≦(1 − ε)2^1/p (f∈L _p ⁺ , ε>0 independent off) implies already lim_n n→∞ ‖T ⁿf −T ⁿ⁺¹ n+1f ‖_p p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown.     

The riesz turndown collar theorem giving an asymptotic estimate of the powers of an operator under the ritt condition

For Banach space operatorsT satisfying the Tadmor-Ritt condition ‖(zI−T)⁻¹‖≤C|z−1|⁻¹, |z|>1, we show how to use the Riesz turndown collar theorem to estimate sup _n≥0‖T ⁿ‖. A similar estimate is shown for lim sup _n ‖T ⁿ‖ in terms of the Ritt constantM=lim sup _z→1‖(1−z)(zI−T)⁻¹‖. We also obtain an estimate of the functional calculus for these operators proving, in particular, that ‖f(T)‖≤C _q‖f‖ _Mult , where ‖·‖ _Mult stands for the multiplier norm of the Cauchy-Stieltjes integrals over a Lusin type cone domain depending onC and a parameterq, 0<q<1. Notation.D denotes the open unit disc of the complex plane,D={z∈ℂ:|z|<1}, andT={z∈ℂ:|z|=1} is the unit circle.H ^∞ is the Banach algebra of bounded analytic functions onD equipped with the supremum norm ‖.‖_∞.     

Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type

LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup _i→∞{sup _y∈K ‖t ⁱx−Tⁱy∼−‖x−y‖}≦0. IfT ^N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tⁱx−Tⁱy‖≦k _i‖x−y∼,x,y∈K, wherek _i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ₀ (X), be less than one. Research supported by National Science Foundation Grant GP 18045.     

Coboundaries of irreducible markov operators onC(K)

LetK be a compact Hausdorff space, and letT be an irreducible Markov operator onC(K). We show that ifgεC(K) satisfies sup _N ‖Σ _j ^=0N T ^j g‖<∞, then (and only then) there existsfεC(K) with (I − T)f=g. Generalizing the result to irreducible Markov operator representations of certain semi-groups, we obtain that bounded cocycles are (continuous) coboundaries. For minimal semi-group actions inC(K), no restriction on the semi-group is needed.     

On the basis property of root vectors of a perturbed self-adjoint operator

We study perturbations of a self-adjoint operator T with discrete spectrum such that the number of its points on any unit-length interval of the real axis is uniformly bounded. We prove that if ‖Bϕ _n ‖ ≤ const, where ϕ _n is an orthonormal system of eigenvectors of the operator T, then the system of root vectors of the perturbed operator T + B forms a basis with parentheses. We also prove that the eigenvalue-counting functions of T and T + B satisfy the relation |n(r, T) − n(r, T + B)| ≤ const.     

Estimates of the periods of periodic points for nonexpansive operators

Suppose thatE is a finite-dimensional Banach space with a polyhedral norm ‖·‖, i.e., a norm such that the unit ball inE is a polyhedron. ℝ ⁿ with the sup norm or ℝ ⁿ with thel ₁-norm are important examples. IfD is a bounded set inE andT:D→D is a map such that ‖T(y)−T(z)‖≤ ‖y−z‖ for ally andz inE, thenT is called nonexpansive with respect to ‖·‖, and it is known that for eachx ∈D there is an integerp=p(x) such that lim _j→∞ T ^jp (x) exists. Furthermore, there exists an integerN, depending only on the dimension ofE and the polyhedral norm onE, such thatp(x)≤N: see [1,12,18,19] and the references to the literature there. In [15], Scheutzow has raised a question about the optimal choice ofN whenE=ℝ ⁿ ,D=K ⁿ , the set of nonnegative vectors in ℝ ⁿ , and the norm is thel ₁-norm. We provide here a reasonably sharp answer to Scheutzow’s question, and in fact we provide a systematic way to generate examples and use this approach to prove that our estimates are optimal forn≤24. See Theorem 2.1, Table 2.1 and the examples in Section 3. As we show in Corollary 2.3, these results also provide information about the caseD=ℝ ⁿ , i.e.,T:ℝ ⁿ →ℝ ⁿ isl ₁-nonexpansive. In addition, it is conjectured in [12] thatN=2 ⁿ whenE=ℝ ⁿ and the norm is the sup norm, and such a result is optimal, if true. Our theorems here show that a sharper result is true for an important subclass of nonexpansive mapsT:(ℝ ⁿ ,‖ · ‖_∞)→(ℝ ⁿ ,‖ · ‖_∞). Partially supported by NSF DMS89-03018.     

Embeddings ofl p m intol ∞ n , 1≦p≦2

Isomorphic embeddings ofl _l ^m intol _∞ ⁿ are studied, and ford(n, k)=inf{‖T ‖ ‖T ⁻¹ ‖;T varies over all isomorphic embeddings ofl ₁ ^[klog2n] intol _∞ ⁿ we have that lim _n→∞ d(n, k)=γ(k)⁻¹,k>1, whereγ(k) is the solution of (1+γ)ln(1+γ)+(1 −γ)ln(1 −γ)=k ⁻¹ln4. Here [x] denotes the integer part of the real numberx.     

p-Energia in spazi metrici generalizzati

Given a setS and a function σ:S x S→[0, +∞[ such that σ(x, x)=0, we define the generalizedp-energy of a curve γ: [a, b]→S, so that, ifS is a Hilbert space and σ(x, y)=‖x−y‖ we obtain . Sufficient conditions for the equality of the defined energies are also given. Moreover the case in whichS is a set of measurable parts of ℝⁿ and σ_r is a family of functions used in order to study the generalized minimizing motions, is discussed.

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Conferenza tenuta il 30 ottobre 1995     

Polynomials of best approximation inC[−1,1]

Equivalences between the condition |P _n ^(k) (x)|≦K(n ⁻¹√1−x ²+1/n ²) ^k n ^-a, whereP _n(x) is the bestn-th degree polynomial approximation tof(x), and the Peetre interpolation space betweenC[−1,1] and the space (1−x ²) ^k f ^(2k)(x)∈C[−1,1] is established. A similar result is shown forE _n(f)= ‖f−P _n‖ _C[−1,1]. Rates other thann ^-a are also discussed. Supported by NSERC grant A4816 of Canada.     

An ergodic theorem on Banach lattices

E is a Banach lattice that is weakly sequentially complete and has a weak unitu. TLf _n=ϕ means that the infimum of |f _n−ϕ| andu converges strongly to zero.T is a positive contraction operator onE andA _n=(1/n)(I+T+...+T ⁿ⁻¹). Without an additional assumption onE, the “truncated limit” TLA _nf need not exist forf inE. This limit exists for eachf ifE satisfies the following additional assumption (C): For everyf inE ₊ and for every numberα>0, there is a numberβ=β(f, α) such that ifg is inE ₊, ‖g‖≦1, 0≦f′≦f and ‖f′‖>α then ‖f′+g‖≧‖g‖+β. Research of this author is partially supported by NSERC Grant A3974. Research of this author is partially supported by NSF Grant 8301619.     

Difference estimates for continuous and discrete operator semigroups

For suitable bounded operator semigroups (e ^tA ) _t≥0 in a Banach space, we characterize the estimate ‖Ae ^tA ‖≤c/F(t) for large t, where F is a function satisfying a sublinear growth condition. The characterizations are by holomorphy estimates on the semigroup, and by estimates on powers of the resolvent. We give similar characterizations of the difference estimate ‖T ⁿ −T ⁿ⁺¹‖≤c/F(n) for a power-bounded linear operator T, when F(n) grows faster than n ^1/2 for large n.     

Lipschitz selections of set-valued mappings and Helly’s theorem

We prove a Helly-type theorem for the family of all m-dimensional convex compact subsets of a Banach space X. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M, ρ) into this family. Let M be finite and let F be such a mapping satisfying the following condition: for every subset M′ ⊂ M consisting of at most 2^m+1 points, the restriction F|_M′ of F to M′ has a selection f_M′ (i. e., f_M′(x) ∈ F(x) for all x ∈ M′) satisfying the Lipschitz condition ‖ƒ_M′(x) − ƒ_M′(y)‖_X ≤ ρ(x, y), x, y ∈ M′. Then F has a Lipschitz selection ƒ: M → X such that ‖ƒ(x) − ƒ(y)‖_X ≤ γρ(x,y), x, y ∈ M where γ is a constant depending only on m and the cardinality of M. We prove that in general, the upper bound of the number of points in M′, 2^m+1, is sharp. If dim X = 2, then the result is true for arbitrary (not necessarily finite) metric space. We apply this result to Whitney’s extension problem for spaces of smooth functions. In particular, we obtain a constructive necessary and sufficient condition for a function defined on a closed subset of R ² to be the restriction of a function from the Sobolev space W _∞ ² (R ²).A similar result is proved for the space of functions on R ² satisfying the Zygmund condition.     

ON NUMBER OF LEAVES ANDBANDWIDTH OF TREES

1.IntroductionFOragraphG=(V,E)oforderp,aonetoonemappingfromVinto{l,2,',p}iscalledanumberingofG.Definition1.1.SupposefisanumberingofG.LetBj(G)=(u57teif(u)--f(v)l.ThebandwidthofG,denotedbyB(G),isdefinedbyB(G)=min{Bf(G)IfisanumberingofG}.Thebandwidthproblemofgraphshasbecomeveryimportantsincethemid-sixties(see[21or[4]).Itisverydifficulttodeterminethebandwidthofagraph.GareyetallllshowedthatthebandwidthproblemisNP-completeevenifitisrestrictedtotreeswithmaximumdegree3.Soitisinterestingtoe…     

On a criterion for the uniform boundedness of a <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup of operators in a Hilbert space

Let T(t), t ≥ 0, be a C ₀-semigroup of linear operators acting in a Hilbert space H with norm ‖·‖. We prove that T(t) is uniformly bounded, i.e., ‖T(t)‖ ≤ M, t ≥ 0, if and only if the following condition is satisfied:

Real Paley-Wiener type theorems for the Dunkl transform on {\mathcal{S}}'(\mathbb{R}^{d})

We prove real Paley-Wiener type theorems for the Dunkl transform ℱ _D on the space of tempered distributions. Let T∈S′(ℝ ^d ) and Δ _κ the Dunkl Laplacian operator. First, we establish that the support of ℱ _D (T) is included in the Euclidean ball , M>0, if and only if for all R>M we have lim  _n→+∞ R ⁻²ⁿ Δ _κ ⁿ T=0 in S′(ℝ ^d ). Second, we prove that the support of ℱ _D (T) is included in ℝ ^d ∖B(0,M), M>0, if and only if for all R<M, we have lim  _n→+∞ R ²ⁿ  ℱ _D ⁻¹(‖y‖⁻²ⁿ ℱ _D (T))=0 in S′(ℝ ^d ). Finally, we study real Paley-Wiener theorems associated with -slowly increasing function.      

Some remarks on Legendrian rectifiable currents

A rectifiable current of dimension n−1 in the sphere bundle Sℝⁿ≃ℝⁿ×S ^{n −1} for euclidean space is Legendrian if it annihilates the contact 1-form α (i.e. T(α∧φ)=0 for all forms φ of degree n−2). Such a current may be naturally associated to any convex set or to any singular real analytic variety, and induces the curvature measures of such a set. We prove that the projection to ℝⁿ of a carrier of a general such T is C ²-rectifiable in the sense of Anzellotti–Serapioni. We deduce that the boundary of a set with positive reach, as well as its singular skeleta, are C ²-rectifiable. In case ∂T= 0 we prove also that the curvature measures associated to T satisfy the analogues of the classical variational formulas for curvature integrals. It follows that such formulas are valid for the curvature measures of subsets of space forms. Received: 3 December 1997/ Revised version: 25 May 1998     

Eigenvalues and power growth

LetX be a complex Banach space and letT be a bounded linear operator onX. Denote by σ _p (T) the point spectrum ofT and by the unit circle. We investigate how the growth of the sequence ‖T ⁿ ‖ is influenced by the size of the set (T) and by the geometry of the spaceX. We also prove analogous results forC ₀-semigroups(T _t )_t≥0. Research partially supported by grants from NSERC, FQRNT and the Canada research chairs program.     

Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone inR ⁿ and −Γ′ be the antipodes of the dual cone of Γ. Let be a partial differential operator with constant coefficients inR ⁿ, whereQ(ζ)≠0 onR ⁿ−iΓ′ andP _i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R ⁿ−iΓ′;P _j(ζ)=0, gradP _j(ζ)≠0} contains some real point on which gradP _j≠0 and gradP _j∉Γ∪(−Γ). LetC be an open cone inR ⁿ−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R ⁿ;P(ξ)=0}. Ifu∈ℒ′∩L _loc ² (R ⁿ−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition implies that the support ofu is contained in Γ.     

Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces

Let （X, Xn; n ≥1） be a sequence of i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with covariance operator ∑. Set Sn = X1 ＋ X2 ＋ ... ＋ Xn, n≥ 1. We prove that, for b 〉 -1,
lim ε→0 ε^2（b＋1） ∞ ∑n=1 （logn）^b/n^3/2 E{｜｜Sn｜｜-σε√nlogn}=σ^-2（b＋1）/（2b＋3）（b＋1） B｜｜Y｜^2b＋3
holds if EX=0,and E｜｜X｜｜^2（log｜｜x｜｜）^3bv（b＋4）〈∞ where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ∑, and σ^2 denotes the largest eigenvalue of ∑.     

On the ratio of the expected maximum of a martingale and theL p-norm of its last term

For eachp>1, the supremum,S, of the absolute value of a martingale terminating at a random variableX inL _p, satisfiesES≦(Γ(q))^1/q‖X‖_p (q=p(p-1)^-1).The maximum,M, of a mean-zero martingale which starts at zero and terminates atX, satisfiesES≦(Γ(q))^1/q‖X‖_p (q=p(p-1)^-1), whereσ _q is the unique solution of the equationt = ‖Z −t ‖ _q for an exponentially distributed random variableZ with mean 1.σ _p has other characterizations and satisfies lim _p‖ q ^{− 1} σ _q =c withc determined byce ^c+1 = 1. Equalities in (1) and (2) are attainable by appropriate martingales which can be realized as stopped segments of Brownian motion. A presumably new property of the exponential distribution is obtained en route to inequality (2).