Selfadjoint operators, normal operators, and characterizations (original) (raw)
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Unbounded operators having self-adjoint or normal powers and some related results
arXiv (Cornell University), 2020
We show that a densely defined closable operator A such that the resolvent set of A 2 is not empty, is necessarily closed. This result is then extended to the case of a polynomial p(A). We also generalize a recent result by Sebestyén-Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given, one of them being a proof that if T is a quasinormal (unbounded) operator such that T n is normal for some n ≥ 2, then T is normal. By a recent result by Pietrzycki-Stochel, we infer that a closed subnormal operator such that T n is normal, must be normal. Another remarkable result is the fact that a hyponormal operator A, bounded or not, such that A p and A q are self-adjoint for some co-prime numbers p and q, is self-adjoint. It is also shown that an invertible operator (bounded or not) A for which A p and A q are normal for some co-prime numbers p and q, is normal. These two results are shown using Bézout's theorem in arithmetic. Notation First, we assume that readers have some familiarity with the standard notions and results in operator theory (see e.g. [17] and [25] for some background). We do recall most of the needed notions though. First, note that in this paper all operators are linear. Let H be a complex Hilbert space and let B(H) be the algebra of all bounded linear operators defined from H into H. If S and T are two linear operators with domains D(S) ⊂ H and D(T) ⊂ H respectively, then T is said to be an extension of S, written S ⊂ T , when D(S) ⊂ D(T) and S and T coincide on D(S). The product ST and the sum S + T of two operators S and T are defined in the usual fashion on the natural domains: D(ST) = {x ∈ D(T) : T x ∈ D(S)} and D(S + T) = D(S) ∩ D(T).
A Criterion for the Normality of Unbounded Operators and Applications to Self-adjointness
arXiv (Cornell University), 2013
In this paper we give and prove a criterion for the normality of unbounded closed operators, which is a sort of a maximality result which will be called "double maximality". As applications, we show, under some assumptions, that the sum of two symmetric operators is essentially self-adjoint; and that the sum of two unbounded normal operators is essentially normal. Some other important results are also established.
Some Conditions on Non-Normal Operators which Imply Normality
2012
In this paper, we prove the following assertions: (i) Let A,B, X ∈ B(H) be such that A∗ is p-hyponormal or log-hyponormal, B is a dominant and X is invertible. If XA = BX, then there is a unitary operator U such that AU = UB and hence A and B are normal. (ii) Let T = A + iB ∈ B(H) be the cartesian decomposition of T with AB is p-hyponormal. If A or B is positive, then T is normal. (iii) Let A, V, X ∈ B(H) be such that V,X are isometries and A∗ is p-hyponormal. If V X = XA, then A is unitary. (iv) Let A,B ∈ B(H) be such that A + B ≥ ±X. Then for every paranormal operator X ∈ B(H) we have ‖AX + XB‖ ≥ ‖X‖2.
SELF-ADJOINT OPERATORS AFFILIATED TO C*-ALGEBRAS
Reviews in Mathematical Physics, 2004
We discuss criteria for the affiliation of a self-adjoint operator to a C * -algebra. We consider in particular the case of graded C * -algebras and we give applications to Hamiltonians describing the motion of dispersive N -body systems and the wave propagation in pluristratified media.
Commutativity theorems for normaloid Hilbert space operators
Journal of Mathematical Analysis and Applications, 2015
The pair (A, B) satisfies (the Putnam-Fuglede) commutativity property δ, respectively , if δ −1 AB (0) ⊆ δ −1 A * B * (0), respectively (AB − 1) −1 (0) ⊆ (A * B * − 1) −1 (0). Normaloid operators do not satisfy either of the properties δ or. This paper considers commutativity properties (δ A,λB) −1 (0) ⊆ (δ A * ,λB *) −1 (0) and (A,B − λ) −1 (0) ⊆ (A * ,B * − λ) −1 (0) for some choices of scalars λ and normaloid operators A, B. Starting with normaloid A, B ∈ B(H) such that the isolated points of their spectrum are normal eigenvalues of the operator, we prove that: (a) if (0 =)λ ∈ isoσ(L A R B) then (A,B − λ) −1 (0) ⊆ (A * ,B * − λ) −1 (0); (b) if 0 / ∈ σ p (A) ∩ σ p (B *) and 0 ∈ isoσ(L A − R λB) then (δ A,λB) −1 (0) ⊆ (δ A * ,λB *) −1 (0). Let σ π (T) denote the peripheral spectrum of the operator T. If A, B are normaloid, then: (i) either dim(B(H)/(A,B − λ)(B(H))) = ∞ for all λ ∈ σ π (A,B), or, there exists a λ ∈ σ π (A,B) ∩ σ p (A,B); (ii) if X is Hilbert-Schmidt, and AXB − λX = 0 for some λ ∈ σ π (A,B), then A * XB * − λX = 0; (iii) if V * ∈ B(H) is an isometry, λ ∈ σ π (A), A −1 (0) ⊆ A * −1 (0), and AXV − λX = 0 (or, AX − λXV = 0) for some X ∈ B(H), then A * XV * − λX = 0 (resp., A * X − λXV * = 0).
On typical properties of Hilbert space operators
Israel Journal of Mathematics, 2012
We study the typical behavior of bounded linear operators on infinite dimensional complex separable Hilbert spaces in the norm, strong-star, strong, weak polynomial and weak topologies. In particular, we investigate typical spectral properties, the problem of unitary equivalence of typical operators, and their embeddability into C 0semigroups. Our results provide information on the applicability of Baire category methods in the theory of Hilbert space operators.
When Products Of Selfadjoints Are Normal
Proceedings of the American Mathematical Society
. Suppose that h; k 2 L(H) are two selfadjoint bounded operators on a Hilbert space H. It is elementary to show that hk is selfadjoint precisely when hk = kh. We answer the following question: under what circumstances must hk be selfadjoint given that it is normal? Throughout A will be a complex unital Banach algebra with unit e. Recall that an element a 2 A is hermitian if kexp(ita)k = 1 (t 2 R) and hermitian-equivalent if sup t2R kexp(ita)k ! 1: and that n is normal (-equivalent) if n = r + is where rs = sr and r; s are hermitian (-equivalent): we write n = r Gamma is for such an n. We shall use an extended two-normal Fuglede theorem: Theorem. Let A be a complex unital Banach algebra. If m; n (2 A) are normalequivalent, and na = am for some a 2 A, then n a = am . This follows from the fact that for any complex number we have a = e i n a e Gammai m so that 7! e in a e Gammaim = e i[ n+n ] a e Gammai[ m+m ] is bounded and analytic, hence c...
Some conditions implying normality of operators
Comptes Rendus Mathematique, 2011
Let T ∈ B(H) and T = U |T | be its polar decomposition. We proved that (i) if T is log-hyponormal or p-hyponormal and U n = U * for some n, then T is normal; (ii) if the spectrum of U is contained in some open semicircle, then T is normal if and only if so is its Aluthge transform T = |T |
Revista Colombiana de Matemáticas, 2005
Abstract. In this paper we will investigate the normality in (WN) and (Y) classes. Keywords and phrases. Normal operators, Hilbert space, hermitian operators. 2000 Mathematics Subject Classification. Primary: 47A15. Secondary: 47B20, 47A63. ... Resumen. En este artıculo ...
Some equivalence classes of operators on B(H)
Bulletin of The Iranian Mathematical Society, 2011
Let L(B(H)) be the algebra of all linear operators on B(H) and P be a property on B(H). For 1, 2 2 L(B(H)), we say that 1 P 2, whenever 1(T) has property P, if and only if 2(T) has this property. In particular, if I is the identity map on B(H), then PI means that preserves property P in both directions. Each property P produces an equivalence relation on L(B(H)). We study the relation between equivalence classes with respect to dif- ferent properties such as being Fredholm, semi-Fredholm, compact, finite rank, generalized invertible, or having a specific semi-index. Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. We denote by F(H) and K(H) the ideals of all finite rank and compact operators in B(H), respectively. The Calkin algebra of H is the quotient algebra C(H) = B(H)/K(H). An operator T 2 B(H) is said to be a Fredholm operator if Im(T), the range of T, is closed and both its kernel and co-kernel are fin...
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