Extension of isometries on the unit sphere of L p spaces

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of Lp (μ) (1 p ∞, p≠2) and a Banach space E can be extended to a linear isometry from Lp(μ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of Lp(μ), then E is linearly isometric to Lp(μ). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of Lp (μ1, H1) and Lp(μ2,H2) must be an isometry and can be extended to a linear isometry from Lp (μ1,H1) to Lp (μ2,H2), where H1 and H2 are Hilbert spaces.     

On the similarity of linear operators in L p to integral operators of the first and second kind

We construct an example of a compact operator of the third kind in L _p (p ≠ 2) not similar to any integral operator of the first or second kind. This example shows that not every linear equation of the third kind in L _p (p ≠ 2) can be reduced by an invertible continuous linear change to an equivalent integral equation of the first or second kind. The example also proves the impossibility of a characterization of integral and Carleman integral operators in L _p (p ≠ 2) in terms of the spectrum and its components.     

A characterization of the linear groups L 2(p)

Let G be a finite group and π _e (G) be the set of element orders of G. Let k ∈ π _e (G) and m _k be the number of elements of order k in G. Set nse(G):= {m _k : k ∈ π _e (G)}. In fact nse(G) is the set of sizes of elements with the same order in G. In this paper, by nse(G) and order, we give a new characterization of finite projective special linear groups L ₂(p) over a field with p elements, where p is prime. We prove the following theorem: If G is a group such that |G| = |L ₂(p)| and nse(G) consists of 1, p ² ? 1, p(p + ?)/2 and some numbers divisible by 2p, where p is a prime greater than 3 with p ≡ 1 modulo 4, then G ? L ₂(p).     

The failure of the Hardy inequality and interpolation of intersections

The main idea of this paper is to clarify why it is sometimes incorrect to interpolate inequalities in a “formal” way. For this we consider two Hardy type inequalities, which are true for each parameter α≠0 but which fail for the “critical” point α=0. This means that we cannot interpolate these inequalities between the noncritical points α=1 and α=?1 and conclude that it is also true at the critical point α=0. Why? An accurate analysis shows that this problem is connected with the investigation of the interpolation of intersections (N∩L _p(w₀), N∩L_p(w₁)), whereN is the linear space which consists of all functions with the integral equal to 0. We calculate theK-functional for the couple (N∩L _p(w₀),N∩L _p (w₁)), which turns out to be essentially different from theK-functional for (L _p(w₀), L_p(w₁)), even for the case whenN∩L _p(w_i) is dense inL _p(w_i) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.     

Recognition of the groups <Emphasis Type="Italic">E</Emphasis><Subscript>7</Subscript>(2) and <Emphasis Type="Italic">E</Emphasis><Subscript>7</Subscript>(3) by prime graph

It is proved that the groups E ₇(2) and E ₇(3) are recognizable by their prime graphs. As a corollary, this completes the proof of V.D. Mazurov’s conjecture that every finite simple group whose prime graph has at least three connected components is either recognizable by spectrum or isomorphic to A ₆.     

Absence of eigenvalues for the periodic Schrödinger operator with singular potential in a rectangular cylinder

We consider the periodic Schrödinger operator on a d-dimensional cylinder with rectangular section. The electric potential may contain a singular component of the form σ(x, y)δ _Σ(x,y), where Σ is a periodic system of hypersurfaces. We establish that there are no eigenvalues in the spectrum of this operator, provided that Σ is sufficiently smooth and σ ∈ L _p,loc(Σ), p > d ? 1.     

Compact elements of the algebras PM p (G)and PF p (G) of a locally compact group

Compact and weakly compact elements of the group algebra L ¹ (G) of a locally compact group G, have been considered by a number of authors. In these investigations it has been shown that, if G is non-compact, then the only weakly compact element of L ¹ (G ) is zero. Conversely, if G is compact, then every element of L ¹ (G) is compact. For 1<p<∞, let PM _p (G)and PF _p (G) denote the closure of L ¹ (G), considered as an algebra of convolution operators on L ^p (G), with respect to the weak operator topology and the norm topology, respectively, in B(L ^p (G), b), the bounded linear operators on L ¹ (G). We study the question of characterizing compact and weakly compact elements of the algebras PM _p (G)and PF _p (G).     

Recognition of finite groups by a set of orders of their elements

For G a finite group, ω(G) denotes the set of orders of elements in G. If ω is a subset of the set of natural numbers, h(ω) stands for the number of nonisomorphic groups G such that ω(G)=ω. We say that G is recognizable (by ω(G)) if h(ω(G))=1. G is almost recognizable (resp., nonrecognizable) if h(ω(G)) is finite (resp., infinite). It is shown that almost simple groups PGL_n(q) are nonrecognizable for infinitely many pairs (n, q). It is also proved that a simple group S₄(7) is recognizable, whereas A₁₀, U₃(5), U₃(7), U₄(2), and U₅(2) are not. From this, the following theorem is derived. Let G be a finite simple group such that every prime divisor of its order is at most 11. Then one of the following holds: (i) G is isomorphic to A₅, A₇, A₈, A₉, A₁₁, A₁₂, L₂(q), q=7, 8, 11, 49, L₃(4), S₄(7), U₄(3), U₆(2), M₁₁, M₁₂, M₂₂, HS, or M^cL, and G is recognizable by the set ω(G); (ii) G is isomorphic to A₆, A₁₀, U₃(3), U₄(2), U₅(2), U₃(5), or J₂, and G is nonrecognizable; (iii) G is isomorphic to S₆(2) or O ₈ ⁺ (2), and h(ω(G))=2. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 651–666, November–December, 1998.     

Interpolation orbits in couples of Lp spacesOrbites d'interpolation pour les couples d'espaces Lp

We consider linear operators T mapping a couple of weighted L_p spaces {L_p0(U₀), L_p1(U₁)} into {L_q0(V₀),L_q1(V₁)} for any 1?p₀, p₁, q₀, q₁?∞, and describe the interpolation orbit of any a∈L_p0(U₀)+L_p1(U₁) that is we describe a space of all {Ta}, where T runs over all linear bounded mappings from {L_p0(U₀),L_p1(U₁)} into {L_q0(V₀),L_q1(V₁)}. We show that interpolation orbit is obtained by the Lions–Peetre method of means with functional parameter as well as by the K-method with a weighted Orlicz space as a parameter. To cite this article: V.I. Ovchinnikov, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 881–884.     

A dynamical systems result on asymptotic integration of linear differential systems

We are interested in the asymptotic integration of linear differential systems of the form x′=[Λ(t)+R(t)]x, where Λ is diagonal and R∈L^p[t₀,∞) for p∈[1,2]. Our dichotomy condition is in terms of the spectrum of the omega-limit set ω_Λ. Our results include examples that are not covered by the Hartman-Wintner theorem.     

OD-characterization of all finite nonabelian simple groups with orders having prime divisors at most 13

The main purpose of this article is to determine h _OD (M) for every finite nonabelian simple group M with order having prime divisors at most 13. This result is an analog of the result by A. V. Vasil’ev [1] about the recognizability of these simple groups by spectrum (the set of element orders). By the available results, we need only consider the groups L ₆(3), U ₄(5), G ₂(4), L ₅(3), S ₄(8), U ₆(2), and O ₈ ⁺ (3).     

On Recognition by Spectrum of Finite Simple Linear Groups over Fields of Characteristic 2

A finite group G is said to be recognizable by spectrum, i.e., by the set of element orders, if every finite group H having the same spectrum as G is isomorphic to G. We prove that the simple linear groups L _n (2^k) are recognizable by spectrum for n = 2^m ≥ 32.Original Russian Text Copyright © 2005 Vasil’ev A. V. and Grechkoseeva M. A.The authors were supported by the Russian Foundation for Basic Research (Grant 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program “ Development of the Scientific Potential of Higher School” of the Ministry for Education of the Russian Federation (Grant 8294), the Program “Universities of Russia” (Grant UR.04.01.202), and a grant of the Presidium of the Siberian Branch of the Russian Academy of Sciences (No. 86-197).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 749–758, July–August, 2005.     

On finite groups isospectral to simple linear and unitary groups

Let L be a simple linear or unitary group of dimension larger than 3 over a finite field of characteristic p. We deal with the class of finite groups isospectral to L. It is known that a group of this class has a unique nonabelian composition factor. We prove that if L ≠ U ₄(2), U ₅(2) then this factor is isomorphic to either L or a group of Lie type over a field of characteristic different from p.     

Groups with the same prime graph as the orthogonal group B n (3)

Let G be a finite group. The prime graph of G is denoted by Γ(G). It is proved in [1] that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then G ? B _p (3) or C _p (3). In this paper we prove the main result that if G is a finite group such that Γ(G) = Γ(B _n (3)), where n ≥ 6, then G has a unique nonabelian composition factor isomorphic to B _n (3) or C _n (3). Also if Γ(G) = Γ(B ₄(3)), then G has a unique nonabelian composition factor isomorphic to B ₄(3), C ₄(3), or ² D ₄(3). It is proved in [2] that if p is an odd prime, then B _p (3) is recognizable by element orders. We give a corollary of our result, generalize the result of [2], and prove that B _2k+1(3) is recognizable by the set of element orders. Also the quasirecognition of B _2k (3) by the set of element orders is obtained.     

Weighted L p -algebras on groups

The space L _p (G), 1 > p < ∞, on a locally compact group G is known to be closed under convolution only if G is compact. However, the weighted spaces L _p (G, w) are Banach algebras with respect to convolution and natural norm under certain conditions on the weight. In the present paper, sufficient conditions for a weight defining a convolution algebra are stated in general form. These conditions are well known in some special cases. The spectrum (the maximal ideal space) of the algebra L _p (G,w) on an Abelian group G is described. It is shown that all algebras of this type are semisimple.     

Stabilization for equations of one-dimensional viscous compressible heat-conducting media with nonmonotone equation of state

We consider the Navier-Stokes system describing motions of viscous compressible heat-conducting and “self-gravitating” media. We use the state function of the form p(u,θ)=p₀(u)+p₁(u)θ linear with respect to the temperature θ, but we admit rather general nonmonotone functions p₀ and p₁ of u, which allows us to treat various physical models of nuclear fluids (for which p and u are the pressure and the specific volume) or thermoviscoelastic solids. For solutions to an associated initial-boundary value problem with “fixed-free” boundary conditions and arbitrarily large data, we prove a collection of estimates independent of time interval, including uniform two-sided bounds for u, and describe asymptotic behavior as t→∞. Namely, we establish the stabilization pointwisely and in L^q for u, in L² for θ, and in L^q for v (the velocity), for any q∈[2,∞). For completeness, the proof of the corresponding global existence theorem is also included.     

Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces

Let T be an injective bilateral weighted shift onl ² thought as "multiplication by λ" on a space of formal Laurent series L²(β). (a) If L²(β) is contained in a space of quasi-analytic class of functions, then the point spectrum σ_p(T^?) of T^? contains a circle and the cyclic invariant subspaceM _f of T generated by f is simply invariant (i.e., ∩{(T^k M _f)^?: k ≥ 0}= {0}) for each f in L²(β); (b) If L²(β) contains a non-quasi-analytic class of functions (defined on a circle г) of a certain type related with the weight sequence of T, then there exists f in L²(ß) such thatM _f is a non-trivial doubly invariant subspace (i.e., (TM _f)^? =M _f); furthermore, if г ? σ_p(T^*), then σ_p (T^*) = г and f can be chosen so that σ_p([T∣M _f]^*) = г?{α}, for some α ε г. Several examples show that the gap between operators satisfying (a) and operators satisfying (b) is rather small.     

On the signless Laplacian spectral characterization of the line graphs of T-shape trees

A graph is determined by its signless Laplacian spectrum if no other nonisomorphic graph has the same signless Laplacian spectrum (simply G is DQS). Let T (a, b, c) denote the T-shape tree obtained by identifying the end vertices of three paths P _a+2, P _b+2 and P _c+2. We prove that its all line graphs L(T(a, b, c)) except L(T(t, t, 2t+1)) (t ? 1) are DQS, and determine the graphs which have the same signless Laplacian spectrum as L(T(t, t, 2t + 1)). Let µ₁(G) be the maximum signless Laplacian eigenvalue of the graph G. We give the limit of µ₁(L(T(a, b, c))), too.     

Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients

Consider a second-order elliptic partial differential operatorL in divergence form with real, symmetric, bounded measurable coefficients, under Dirichlet or Neumann conditions on the boundary of a strongly Lipschitz domain Ω. Suppose that 1 <p < ∞ and μ > 0. ThenL has a bounded H_∞ functional calculus in L^p(Ω), in the sense that ¦¦f (L +cI)u¦¦_p ≤C sup_¦arλ¦<μ ^¦f¦ ¦‖u¦‖_p for some constantsc andC, and all bounded holomorphic functionsf on the sector ¦ argλ¦ < μ that contains the spectrum ofL +cI. We prove this by showing that the operatorsf(L + cI) are Calderón-Zygmund singular integral operators.