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Displaying similar documents to “Operator fractional-linear transformations: convexity and compactness of image; applications”

On closed sets with convex projections in Hilbert space

Stoyu Barov, Jan J. Dijkstra (2007)

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Let k be a fixed natural number. We show that if C is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all k-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets B the set k ( B ) consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that k ( B ) is precisely the intersection of...

Regular fractional iteration of convex functions

Marek Kuczma (1980)

Annales Polonici Mathematici

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The existence of a unique C 1 solution φ of equation (1) is proved under the condition that f: I → I is convex or concave and of class C 1 in I, 0 < f(x) < x in I*, and f’(x) > 0 in I. Here I = [0, a] or [0, a), 0 < a ≤ ∞, and I* = I 0.

Operators on a Hilbert space similar to a part of the backward shift of multiplicity one

Yoichi Uetake (2001)

Studia Mathematica

Similarity:

Let A: X → X be a bounded operator on a separable complex Hilbert space X with an inner product · , · X . For b, c ∈ X, a weak resolvent of A is the complex function of the form ( I - z A ) - 1 b , c X . We will discuss an equivalent condition, in terms of weak resolvents, for A to be similar to a restriction of the backward shift of multiplicity 1.

Extreme points of the complex binary trilinear ball

Fernando Cobos, Thomas Kühn, Jaak Peetre (2000)

Studia Mathematica

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We characterize all the extreme points of the unit ball in the space of trilinear forms on the Hilbert space 2 . This answers a question posed by R. Grząślewicz and K. John [7], who solved the corresponding problem for the real Hilbert space 2 . As an application we determine the best constant in the inequality between the Hilbert-Schmidt norm and the norm of trilinear forms.

Orlicz boundedness for certain classical operators

E. Harboure, O. Salinas, B. Viviani (2002)

Colloquium Mathematicae

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Let ϕ and ψ be functions defined on [0,∞) taking the value zero at zero and with non-negative continuous derivative. Under very mild extra assumptions we find necessary and sufficient conditions for the fractional maximal operator M Ω α , associated to an open bounded set Ω, to be bounded from the Orlicz space L ψ ( Ω ) into L ϕ ( Ω ) , 0 ≤ α < n. For functions ϕ of finite upper type these results can be extended to the Hilbert transform f̃ on the one-dimensional torus and to the fractional integral operator...

Smoothing a polyhedral convex function via cumulant transformation and homogenization

Alberto Seeger (1997)

Annales Polonici Mathematici

Similarity:

Given a polyhedral convex function g: ℝⁿ → ℝ ∪ +∞, it is always possible to construct a family g t > 0 which converges pointwise to g and such that each gₜ: ℝⁿ → ℝ is convex and infinitely often differentiable. The construction of such a family g t > 0 involves the concept of cumulant transformation and a standard homogenization procedure.

Separated sequences in uniformly convex Banach spaces

J. M. A. M. van Neerven (2005)

Colloquium Mathematicae

Similarity:

We give a characterization of uniformly convex Banach spaces in terms of a uniform version of the Kadec-Klee property. As an application we prove that if (xₙ) is a bounded sequence in a uniformly convex Banach space X which is ε-separated for some 0 < ε ≤ 2, then for all norm one vectors x ∈ X there exists a subsequence ( x n j ) of (xₙ) such that i n f j k | | x - ( x n j - x n k ) | | 1 + δ X ( 2 / 3 ε ) , where δ X is the modulus of convexity of X. From this we deduce that the unit sphere of every infinite-dimensional uniformly convex Banach space...

Convex integration with constraints and applications to phase transitions and partial differential equations

Stefan Müller, Vladimír Šverák (1999)

Journal of the European Mathematical Society

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We study solutions of first order partial differential relations D u K , where u : Ω n m is a Lipschitz map and K is a bounded set in m × n matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of D u and second we replace Gromov’s P −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our...

Equivalence of multi-norms

H. G. Dales, M. Daws, H. L. Pham, P. Ramsden

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The theory of multi-norms was developed by H. G. Dales and M. E. Polyakov in a memoir that was published in Dissertationes Mathematicae. In that memoir, the notion of ’equivalence’ of multi-norms was defined. In the present memoir, we make a systematic study of when various pairs of multi-norms are mutually equivalent. In particular, we study when (p,q)-multi-norms defined on spaces L r ( Ω ) are equivalent, resolving most cases; we have stronger results in the case where r = 2. We also show...

On isometrical extension properties of function spaces

Hisao Kato (2015)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces C ( Q ) and C ( Δ ) , where Q and Δ denote the Hilbert cube [ 0 , 1 ] and a Cantor set, respectively.

Minimal multi-convex projections

Grzegorz Lewicki, Michael Prophet (2007)

Studia Mathematica

Similarity:

We say that a function from X = C L [ 0 , 1 ] is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Πₙ ⊂ X, where Πₙ denotes the space of algebraic polynomials of degree less than or equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed, only for k = n-1 and k = n does such a projection exist. So let us consider instead a more general “shape”...

Regularity of solutions of the fractional porous medium flow

Luis Caffarelli, Fernando Soria, Juan Luis Vázquez (2013)

Journal of the European Mathematical Society

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We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is u t = · ( u ( - Δ ) - s u ) , 0 < s < 1 . The problem is posed in { x n , t } with nonnegative initial data u ( x , 0 ) that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and C α regularity of such weak solutions. Finally, we extend...

Fractional integral operators on B p , λ with Morrey-Campanato norms

Katsuo Matsuoka, Eiichi Nakai (2011)

Banach Center Publications

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We introduce function spaces B p , λ with Morrey-Campanato norms, which unify B p , λ , C M O p , λ and Morrey-Campanato spaces, and prove the boundedness of the fractional integral operator I α on these spaces.

Numerical radius inequalities for Hilbert C * -modules

Sadaf Fakri Moghaddam, Alireza Kamel Mirmostafaee (2022)

Mathematica Bohemica

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We present a new method for studying the numerical radius of bounded operators on Hilbert C * -modules. Our method enables us to obtain some new results and generalize some known theorems for bounded operators on Hilbert spaces to bounded adjointable operators on Hilbert C * -module spaces.

The Young inequality and the Δ₂-condition

Philippe Laurençot (2002)

Colloquium Mathematicae

Similarity:

If φ: [0,∞) → [0,∞) is a convex function with φ(0) = 0 and conjugate function φ*, the inequality x y ε φ ( x ) + C ε φ * ( y ) is shown to hold true for every ε ∈ (0,∞) if and only if φ* satisfies the Δ₂-condition.