Wyniki wyszukiwania

1

Young's (in)equality for compact operators

100%

Larotonda G.

Studia Mathematica

|

2016

|

tom 233

|

nr 2

169-181

EN

If a,b are n × n matrices, T. Ando proved that Young's inequality is valid for their singular values: if p > 1 and 1/p + 1/q = 1, then $λ_{k}(|ab*|) ≤ λ_{k}(1/p |a|^{p} + 1/q |b|^{q})$ for all k. Later, this result was extended to the singular values of a pair of compact operators acting on a Hilbert space by J. Erlijman, D. R. Farenick and R. Zeng. In this paper we prove that if a,b are compact operators, then equality holds in Young's inequality if and only if $|a|^{p} = |b|^{q}$.

2

The rectifiable distance in the unitary Fredholm group

63%

Andruchow E., Larotonda G.

Studia Mathematica

|

2010

|

tom 196

|

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

The group of L²-isometries on H¹₀

51%

Andruchow E., Chiumiento E., Larotonda G.

Studia Mathematica

|

2013

|

tom 217

|

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 $ℬ_{p}(H¹₀)$.

Ograniczanie wyników

3 Studia Mathematica

3 Larotonda G.

2 Andruchow E.

1 Chiumiento E.

1 2016

1 2013

1 2010

Young's (in)equality for compact operators

The rectifiable distance in the unitary Fredholm group

The group of L²-isometries on H¹₀