We have missed one type of spaces generating compatible pairs of isometries in 1, Theorem 5.14. In this note, we describe the additional type of compatible pairs of isometries and supplement the ...results. More precisely, we reformulate and correctly prove Theorem 5.14 and we supplement Theorem 5.15 and Theorem 8.3 which follow directly from Theorem 5.14.
Theorems, definitions, formulas etc. enumerated with section numbers, as above, all refers to the paper 1, while that from the corrigendum are singly enumerated.
Let (Ωi,Σi,μi), i=1,2, be two measure spaces, 1<p,q<∞ with 1p+1q=1, Xi=Lp(Ωi,Σi,μi), Yi=Lq(Ωi,Σi,μi), and let Xi+={f∈Xi:f≥0a.e.} be the positive cone of Xi. In this paper, we first show a weak ...stability formula of a standard ε-isometry F:X1+→X2+: For every x⁎∈Y1+, there exists a unique ϕ∈Y2+ with ‖x⁎‖=‖ϕ‖≡r such that|〈x⁎,u〉−〈ϕ,F(u)〉|≤2rε,for allu∈X1+. Making use of it, we show the following Hyers-Ulam stability of ε-isometry F:X1+→X2+: If F is almost surjective, then there exists a unique additive surjective isometry V:X1+→X2+ (the restriction of a linear surjective isometry between X1 and X2) defined as V(u)=limt→∞F(tu)t for each u∈X1+ so that‖F(u)−V(u)‖≤4ε,for allu∈X1+.
In this paper, we introduce the notion of
-partial isometry for a positive operator A and a nonnegative integer m. This family of operators contains both the class of
-isometries discussed in Sid ...Ahmed and Saddi A-m-isometric operators in semi-Hilbertian spaces. Linear Algebra Appl. 2012;436:3930-3942 and that of m-partial isometries introduced in Saddi and Sid Ahmed m-partial isometries on Hilbert spaces. Int J Funct Anal Oper Theory Appl. 2010;2(1):67-83. First, we give some interesting algebraic properties of
-partial isometries, then we discuss a necessary and sufficient condition for an
-partial isometry to be an
-isometry. Finally, we give some spectral properties of
-partial isometries.
Operator theory on the pentablock Jindal, Abhay; Kumar, Poornendu
Journal of mathematical analysis and applications,
12/2024, Letnik:
540, Številka:
1
Journal Article
Recenzirano
The pentablock, denoted as P, is defined as follows:P={(a21,tr(A),det(A)):A=aij2×2 with ‖A‖<1}. It originated from the work of Agler–Lykova–Young in connection with a particular case of the ...μ-synthesis problem. It is a non-convex, polynomially convex, C-convex, star-like about the origin, and inhomogeneous domain.
This paper deals with operator theory on the pentablock. We study pentablock unitaries and isometries, providing an algebraic characterization of pentablock isometries. En route, we provide the Wold-type decomposition for pentablock isometries, which consists of three parts: the unitary part, the pure part, and a new component. We define this novel component as the quasi-pentablock unitary and provide a functional model for it. Additionally, a model for a class of pure pentablock isometries has been found, along with some examples. Furthermore, a representation resembling the Beurling-Lax-Halmos paradigm has been presented for the invariant subspaces of pentablock pure isometries.
We study a class of projections on Banach spaces that are in the convex hull of n surjective isometries. We apply some results to the particular case of spaces of absolutely continuous functions on a ...compact subset of R and with values in a strictly convex Banach space.
Let M and N be complex unital Jordan-Banach algebras, and let M−1 and N−1 denote the sets of invertible elements in M and N, respectively. Suppose that M⊆M−1 and N⊆N−1 are clopen subsets of M−1 and ...N−1, respectively, which are closed for powers, inverses and products of the form Ua(b). In this paper we prove that for each surjective isometry Δ:M→N there exists a surjective real-linear isometry T0:M→N and an element u0 in the McCrimmon radical of N such that Δ(a)=T0(a)+u0 for all a∈M. Assuming that M and N are unital JB⁎-algebras we establish that for each surjective isometry Δ:M→N the element Δ(1)=u is a unitary element in N and there exist a central projection p∈M and a complex-linear Jordan ⁎-isomorphism J from M onto the u⁎-homotope Nu⁎ such thatΔ(a)=J(p∘a)+J((1−p)∘a⁎), for all a∈M. Under the additional hypothesis that there is a unitary element ω0 in N satisfying Uω0(Δ(1))=1, we show the existence of a central projection p∈M and a complex-linear Jordan ⁎-isomorphism Φ from M onto N such thatΔ(a)=Uw0⁎(Φ(p∘a)+Φ((1−p)∘a⁎)), for all a∈M.
Let Q⊆Rm, K⊆Rn be open sets, p,q∈N, 1≤r<∞ and let E,F be Banach spaces. Denote by C⁎p(Q,E)r the space of all f∈Cp(Q,E) with bounded derivatives of order ≤p, endowed with the ...norm‖f‖=sups∈Q‖(‖∂λf(s)‖E)λ∈Λ‖r, where ‖⋅‖r denotes the ℓr norm on RΛ, Λ={λ:|λ|≤p}. Let T:C⁎p(Q,E)r→C⁎q(K,F)r be a linear surjective isometry. Then m=n and p=q and there are a Cp-diffeomorphism τ:K→Q and Banach space isomorphisms V(t):E→F so thatTf(t)=V(t)f(τ(t)) if f∈C⁎p(Q,E),t∈K. The result holds in a more general setting. The proof establishes a direct link between isometries and biseparating maps.
We study the dynamics of the group of isometries of Lp-spaces. In particular, we study the canonical actions of these groups on the space of δ-isometric embeddings of finite dimensional subspaces of ...Lp(0,1) into itself, and we show that for every real number 1≤p<∞ with p≠4,6,8,… they are ε-transitive provided that δ is small enough. We achieve this by extending the classical equimeasurability principle of Plotkin and Rudin. We define the central notion of a Fraïssé Banach space which underlies these results and of which the known separable examples are the spaces Lp(0,1), p≠4,6,8,… and the Gurarij space. We also give a proof of the Ramsey property of the classes {ℓpn}n, p≠2,∞, viewing it as a multidimensional Borsuk-Ulam statement. We relate this to an arithmetic version of the Dual Ramsey Theorem of Graham and Rothschild as well as to the notion of a spreading vector of Matoušek and Rödl. Finally, we give a version of the Kechris-Pestov-Todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space X with a Ramsey property of the collection of finite dimensional subspaces of X.
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