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.     

The bicompletion of an asymmetric normed linear space

A biBanach space is an asymmetric normed linear space (X,‖·‖) such that the normed linear space (X,‖·‖^s) is a Banach space, where ‖x‖^s= max {‖x‖,‖-x‖} for all x∈X. We prove that each asymmetric normed linear space (X,‖·‖) is isometrically isomorphic to a dense subspace of a biBanach space (Y,‖·‖_Y). Furthermore the space (Y,‖·‖_Y) is unique (up to isometric isomorphism). This revised version was published online in August 2006 with corrections to the Cover Date.     

Orthogonal almost locally minimal projections on ℓ 1 n

A projectionP on a Banach spaceX with ‖P‖≤λ₀ is called almost locally minimal if, for every α>0 small enough, the ballB(P,α) in the space of operatorsL(X) does not contain a projectionQ with ‖Q‖≤‖P‖(1–Dα²), whereD=D(λ₀) is a constant independent of ‖P‖. It is shown that, for everyp≥1 and every compact abelian groupG, every translation invariant projection onL _p(G) is almost locally minimal. Orthogonal projections on ℓ ₁ ⁿ are investigated with respect to some weaker local minimality properties. Participant in Workshop in Linear Analysis and Probability, Texas A&M University, College Station, Texas 1998. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

Quasi-compactness and uniform ergodicity of positive operators

The following converse to the Yosida-Kakutani theorem is proved: IfT is a positive operator on a Banach lattice with ‖T ⁿ‖/n → 0, thenT is quasi-compact if (and only if) the averages of its iterates converge uniformly to a finite-dimensional projection.     

Best monotone approximation and the Hardy-Littlewood maximal function

If f∈L₂[0, 1] and g^*∈L₂[0, 1] is the best non-decreasing approximation to f, then it's shown that ‖f−g^*‖₂=‖f−θ(f)‖₂, where θ(f) denotes the Hardy-Littlewood maximal function of f.     

The classification of injective Banach lattices

LetE be a 1-injective Banach lattice,X any Banach space andT: E ← X a norm bounded linear operator. Then eitherT is an isomorphism on some copy ofl _∞ inE or for all σ > 0 there is φ ∈E ₊ ^′ such that ‖Tu‖≦φ (|u|)+σ ‖u‖ for allu ∈E. We deduce the theorem that: A norm order continuous injective Banach lattice is order isomorphic to an (AL)-space.     

An identity for normal-like operators

Forλεσ(A) (A a bounded linear operator on a Hilbert space) withλ a boundary point of the numerical range, the ‘spectral theory’ forλ is ‘just as ifA were normal’. IfA isnormal-like (the smallest disk containingσ(A) has radiusr=inf _z ‖A − z‖), then also sup {‖Ax‖² − |〈x.Ax〉|²:‖x‖=1}=r ². This research was partially supported by Air Force Contract AF-AFOSR-62-414.     

Uniform boundedness of (<Emphasis Type="Italic">C</Emphasis>, 1) means of Jacobi expansions in weighted sup norms. I (The main theorems and ideas)

Lubinsky and Totik’s decomposition [11] of the Cesàro operators σ _n ^(α,β) of Jacobi expansions is modified to prove uniform boundedness in weighted sup norms, i.e., ‖w ^(a,b) σ _n ^(α,β) ‖_∞ ≦ C‖w ^(a,b) f‖_∞, whenever α,β ≧ −1/2 and a, b are within the square around (α/2 + 1/4, α/2 + 1/4) having a side length of 1. This approach uses only classical results from the theory of orthogonal polynomials and various estimates for the Jacobi weights. The present paper is concerned with the main theorems and ideas, while a second paper [7] provides some necessary estimations.      

Double operator integrals and their estimates in the uniform norm

In this paper, conditions are considered for the existence of the double operator integral ∫∫ ϕ(λ,μ)dE_λTdF_μ, where E_λ, F_μ are the spectral functions of tow self-adjoint operators A, B on a Hilbert space and T is a bounded operator. In principal, the case where A has finite spectrum is studied. Nonlinear estimates of ‖f(A)T-T f(B)‖ in terms of the norm of ‖AT-TB‖ for f∈ Lip 1 are deduced. Also, a formula for the Fréchet derivative is presented. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 148–173. Translated by S. V. Kislyakov.     

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 ∑.     

Large deviations of theL p-norm of a wiener process with drift

In this paper we calculate the exact asymptotics of the probability P{‖w(t)+uct‖ _p >u},u→∞, wherew(t) is the standard Wiener process and ‖x‖ _p is the ordinary norm in the spaceL ^p[0,1],p≥2. The result is obtained on the basis of a general theorem due to the author on the asymptotics of the Gaussian measureP(uD),u→∞, for a Borel setD belonging to a separable Banach space. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 429–436, March, 1999.     

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 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:

General Study of Iterative Processes of <Emphasis Type="Italic">R</Emphasis>-Order at Least Three under Weak Convergence Conditions

We consider a family of Newton-type iterative processes solving nonlinear equations in Banach spaces, that generalizes the usually iterative methods of R-order at least three. The convergence of this family in Banach spaces is usually studied when the second derivative of the operator involved is Lipschitz continuous and bounded. In this paper, we relax the first condition, assuming that ‖F″(x)−F″(y)‖≤ω(‖x−y‖), where ω is a nondecreasing continuous real function. We prove that the different R-orders of convergence that we can obtain depend on the quasihomogeneity of the function ω. We end the paper by applying the study to some nonlinear integral equations. This work was supported by the Ministry of Science and Technology (BFM 2002-00222), the University of La Rioja (API-04/13) and the Government of La Rioja (ACPI 2003/2004).     

A characterization of normal operators

LetA be a bounded linear operator in a Hilbert space. IfA is normal then log ‖e ^Ai u‖ and log ‖e ^A·1 u‖ are convex functions for allu≠0. In this paper we prove that these properties characterize normal operators. Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.     

Combined perturbation bounds: Ⅱ. Polar decompositions

In this paper, we study the perturbation bounds for the polar decomposition A= QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ2r||△Q||2F ≤ ||△A||2F,1/2||△H||2F ≤ ||△A||2F and ||△∑||2F ≤ ||△A||2F, respectively, where ∑ = diag(σ1, σ2,..., σr, 0,..., 0) is the singular value matrix of A and σr denotes the smallest nonzero singular value. Here we present some new combined (asymptotic)perturbation bounds σ2r ||△Q||2F 1/2||△H||2F≤ ||△A||2F and σ2r||△Q||2F ||△∑ ||2F ≤||△A||2F which are optimal for each factor. Some corresponding absolute perturbation bounds are also given.     

Equilateral simplices in a four-dimensional normed space

Let E be a four-dimensional real normed space, let P ⊂ E be a 3-plane, and let x ≥ 3/4 be a positive number. It is proved that there exist four equidistant points A₁, A₂, A₃, A₄ ∈ P and a point A₅ ∈ E such that ‖A₅A_i‖ = x · ‖A₁A₂‖ for is = 1, 2, 3, 4. In particular, E contains an equilateral simplex (with base in P). Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 88–91.     

The intersection of Sylow subgroups in finite groups

In Theorem 1, letting p be a prime, we prove: (1) If G=S_n is a symmetric group of degree n, then G contains two Sylow p-subgroups with trivial intersection iff (p, n) ∉ {(3, 3), (2, 2), (2, 4), (2, 8)}, and (2) If H=A_n is an alternating group of degree n, then H contains two Sylow p-subgroups with trivial intersection iff (p, n) ∉ {(3, 3), (2, 4)}. In Theorem 2, we argue that if G is a finite simple non-Abelian group and p is a prime, then G contains a pair of Sylow p-subgroups with trivial intersection. Also we present the corollary which says that if P is a Sylow subgroup of a finite simple non-Abelian group G, then ‖G‖>‖P‖². Supported by RFFR grants Nos. 93-01-01529, 93-01-01501, and 96-01-01893, and by International Science Foundation and Government of Russia grant RPC300. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 424–432, July–August, 1996.     

New associative and nonassociative Gelfand-Naimark theorems

We prove that, ifA is a complex Banach algebra with a unit 1 and a conjugate-linear vector space involution^* such that 1^*=1 and‖a ^*a‖=‖a^*‖ ‖a‖ for alla inA, and ifdim(A)≥3, thenA is a C^*-algebra. The two-dimensional case is also considered and described.     

The Norm of the Laplace Operator Restriction to a Homogeneous Polynomial Space

We investigate the restriction Δ _r,μ of the Laplace operator Δ onto the space of r-variate homogeneous polynomials F of degree μ. In the uniform norm on the unit ball of ℝ ^r , and with the corresponding operator norm, ‖Δ _r,μ F‖≤‖Δ _r,μ ‖⋅‖F‖ holds, where, for arbitrary F, the ‘constant’ ‖Δ _r,μ ‖ is the best possible. We describe ‖Δ _r,μ ‖ with the help of the family T _μ (σ x), , of scaled Chebyshev polynomials of degree μ. On the interval [−1,+1], they alternate at least (μ−1)-times, as the Zolotarev polynomials do, but they differ from them by their symmetry. We call them Zolotarev polynomials of the second kind, and calculate ‖Δ _r,μ ‖ with their help. We derive upper and lower bounds, as well as the asymptotics for μ→∞. For r≥5 and sufficiently large μ, we just get ‖Δ _r,μ ‖=(r−2)μ(μ−1). However, for 2≤r≤4 or lower values of μ, the result is more complicated. This gives the problem a particular flavor. Some Bessel functions and the φcot φ-expansion are involved.