Wyniki wyszukiwania

1

A nonsmooth exponential

100%

Andruchow E.

Studia Mathematica

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2003

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tom 155

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nr 3

265-271

EN

Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set ${ℳ}_{sa}$ of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator $L_{ξ}$ on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), $exp(ξ) = e^{iL_{ξ}}$, is continuous but not differentiable. The same holds for the Cayley transform $C(ξ) = (L_{ξ} - i)(L_{ξ} + i)^{-1}$. We also show that the unitary group $U_{ℳ} ⊂ L²(ℳ,τ)$ with the strong operator topology is not an embedded submanifold of L²(ℳ,τ), in any way which makes the product (u,w) ↦ uw ($u,w ∈ U_{ℳ}$) a differentiable map.

2

The rectifiable distance in the unitary Fredholm group

64%

Andruchow E., Larotonda G.

Studia Mathematica

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2010

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tom 196

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nr 2

151-178

EN

Let $U_{c}(𝓗)$ = {u: u unitary and u-1 compact} stand for the unitary Fredholm group. We prove the following convexity result. Denote by $d_{∞}$ the rectifiable distance induced by the Finsler metric given by the operator norm in $U_{c}(𝓗)$. If $u₀,u₁,u ∈ U_{c}(𝓗)$ and the geodesic β joining u₀ and u₁ in $U_{c}(𝓗)$ satisfy $d_{∞}(u,β) < π/2$, then the map $f(s) = d_{∞}(u,β(s))$ is convex for s ∈ [0,1]. In particular, the convexity radius of the geodesic balls in $U_{c}(𝓗)$ is π/4. The same convexity property holds in the p-Schatten unitary groups $U_{p}(𝓗)$ = {u: u unitary and u-1 in the p-Schatten class} for p an even integer, p ≥ 4 (in this case, the distance is strictly convex). The same results hold in the unitary group of a C*-algebra with a faithful finite trace. We apply this convexity result to establish the existence of curves of minimal length with given initial conditions, in the unitary orbit of an operator, under the action of the Fredholm group. We characterize self-adjoint operators A such that this orbit is a submanifold (of the affine space A + 𝓚(𝓗), where 𝓚(𝓗) = compact operators).

3

Metrics in the sphere of a C*-module

64%

Andruchow E., Varela A.

Open Mathematics

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2007

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tom 5

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nr 4

639-653

EN

Given a unital C*-algebra $$\mathcal{A}$$ and a right C*-module $$\mathcal{X}$$ over $$\mathcal{A}$$ , we consider the problem of finding short smooth curves in the sphere $$\mathcal{S}_\mathcal{X} $$ = {x ∈ $$\mathcal{X}$$ : 〈x, x〉 = 1}. Curves in $$\mathcal{S}_\mathcal{X} $$ are measured considering the Finsler metric which consists of the norm of $$\mathcal{X}$$ at each tangent space of $$\mathcal{S}_\mathcal{X} $$ . The initial value problem is solved, for the case when $$\mathcal{A}$$ is a von Neumann algebra and $$\mathcal{X}$$ is selfdual: for any element x 0 ∈ $$\mathcal{S}_\mathcal{X} $$ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ $$\mathcal{L}_\mathcal{A} (\mathcal{X})$$ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and $$\dot \gamma $$ (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ $$\mathcal{S}_\mathcal{X} $$ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 $$\mathcal{L}_\mathcal{A} (\mathcal{X})$$ f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.

4

The group of L²-isometries on H¹₀

51%

Andruchow E., Chiumiento E., Larotonda G.

Studia Mathematica

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2013

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tom 217

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nr 3

193-217

EN

Let Ω be an open subset of ℝⁿ. Let L² = L²(Ω,dx) and H¹₀ = H¹₀(Ω) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group 𝔾 of invertible operators on H¹₀ which preserve the L²-inner product. When Ω is bounded and ∂Ω is smooth, this group acts as the intertwiner of the H¹₀ solutions of the non-homogeneous Helmholtz equation u - Δu = f, $u|_{∂Ω} = 0$. We show that 𝔾 is a real Banach-Lie group, whose Lie algebra is (i times) the space of symmetrizable operators. We discuss the spectrum of operators belonging to 𝔾 by means of examples. In particular, we give an example of an operator in 𝔾 whose spectrum is not contained in the unit circle. We also study the one-parameter subgroups of 𝔾. Curves of minimal length in 𝔾 are considered. We introduce the subgroups $𝔾_{p}: = 𝔾 ∩ (I - ℬ_{p}(H¹₀))$, where $ℬ_{p}(H₀¹)$ is the Schatten ideal of operators on H₀¹. An invariant (weak) Finsler metric is defined by the p-norm of the Schatten ideal of operators on L². We prove that any pair of operators $G₁, G₂ ∈ 𝔾_{p}$ can be joined by a minimal curve of the form $δ(t) = G₁ e^{itX}$, where X is a symmetrizable operator in $ℬ_{p}(H¹₀)$.

5

Proper subspaces and compatibility

51%

Andruchow E., Chiumiento E., Di Iorio y Lucero M.

Studia Mathematica

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2015

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tom 231

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nr 3

195-218

EN

Let 𝓔 be a Banach space contained in a Hilbert space 𝓛. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambickiĭ, we say that a bounded operator on 𝓔 is a proper operator if it admits an adjoint with respect to the inner product of 𝓛. A proper operator which is self-adjoint with respect to the inner product of 𝓛 is called symmetrizable. By a proper subspace 𝓢 we mean a closed subspace of 𝓔 which is the range of a proper projection. Furthermore, if there exists a symmetrizable projection onto 𝓢, then 𝓢 belongs to a well-known class of subspaces called compatible subspaces. We find equivalent conditions to describe proper subspaces. Then we prove a necessary and sufficient condition for a proper subspace to be compatible. The existence of non-compatible proper subspaces is related to spectral properties of symmetrizable operators. Each proper subspace 𝓢 has a supplement 𝒯 which is also a proper subspace. We give a characterization of the compatibility of both subspaces 𝓢 and 𝒯. Several examples are provided that illustrate different situations between proper and compatible subspaces.

6

A note on the differentiable structure of generalized idempotents

51%

Andruchow E., Corach G., Mbekhta M.

Open Mathematics

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2013

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tom 11

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nr 6

1004-1019

EN

For a fixed n > 2, we study the set Λ of generalized idempotents, which are operators satisfying T n+1 = T. Also the subsets Λ†, of operators such that T n−1 is the Moore-Penrose pseudo-inverse of T, and Λ*, of operators such that T n−1 = T* (known as generalized projections) are studied. The local smooth structure of these sets is examined.

Ograniczanie wyników

4 Studia Mathematica

2 Open Mathematics

6 Andruchow E.

2 Chiumiento E.

2 Larotonda G.

1 Corach G.

1 Di Iorio y Lucero M.

1 Mbekhta M.

1 Varela A.

1 2015

2 2013

1 2010

1 2007

1 2003

A nonsmooth exponential

The rectifiable distance in the unitary Fredholm group

Metrics in the sphere of a C*-module

The group of L²-isometries on H¹₀

Proper subspaces and compatibility

A note on the differentiable structure of generalized idempotents