Unbounded Normal Operators Research Articles

Similarities involving unbounded normal operators

Similarities involving unbounded normal operators

Perturbation of complex polynomials and normal operators

AbstractWe study the regularity of the roots of complex monic polynomials P (t) of fixed degree depending smoothly on a real parameter t. We prove that each continuous parameterization of the roots of a generic C∞ curve P (t) (which always exists) is locally absolutely continuous. Generic means that no two of the continuously chosen roots meet of infinite order of flatness. Simple examples show that one cannot expect a better regularity than absolute continuity. This result will follow from the proposition that for any t0 there exists a positive integer N such that t ↦ P (t0 ± (t – t0)N ) admits smooth parameterizations of its roots near t0. We show that Cn curves P (t) (where n = deg P) admit differentiable roots if and only if the order of contact of the roots is ≥ 1. We give applications to the perturbation theory of normal matrices and unbounded normal operators with compact resolvents and common domain of definition: The eigenvalues and eigenvectors of a generic C∞ curve of such operators can be arranged locally in an absolutely continuous way (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Riesz bases for p-subordinate perturbations of normal operators

For a class of unbounded perturbations of unbounded normal operators, the change of the spectrum is studied and spectral criteria for the existence of a Riesz basis with parentheses of root vectors are established. A Riesz basis without parentheses is obtained under an additional a priori assumption on the spectrum of the perturbed operator. The results are applied to two classes of block operator matrices.

Multicyclicity of unbounded normal operators and polynomial approximation in C

A remarkable and much cited result of Bram [J. Bram, Subnormal operators, Duke Math. J. 22 (1955) 75–94] shows that a star-cyclic bounded normal operator in a separable Hilbert space has a cyclic vector. If, in addition, the operator is multiplication by the variable in a space L 2 ( m ) (not only unitarily equivalent to it), then it has a cyclic vector in L ∞ ( m ) . We extend Bram's result to the case of a general unbounded normal operator, implying by this that the (classical) multiplicity and the multicyclicity of the operator (cf. [N.K. Nikolski, Operators, Functions and Systems: An Easy Reading, vol. 2, Math. Surveys Monogr., vol. 93, Amer. Math. Soc., Providence, 2002]) coincide. It follows that if m is a sigma-finite Borel measure on C (possibly with noncompact support), then there is a nonnegative finite Borel measure τ equivalent to m and such that L 2 ( C , τ ) is the norm-closure of the polynomials in z.

Positive sesquilinear form measures and generalized eigenvalue expansions

Positive operator measures (with values in the space of bounded operators on a Hilbert space) and their generalizations, mainly positive sesquilinear form measures, are considered with the aim of providing a framework for their generalized eigenvalue type expansions. Though there are formal similarities with earlier approaches to special cases of the problem, the paper differs e.g. from standard rigged Hilbert space constructions and avoids the introduction of nuclear spaces. The techniques are predominantly measure theoretic and hence the Hilbert spaces involved are separable. The results range from a Naimark type dilation result to direct integral representations and a fairly concrete generalized eigenvalue expansion for unbounded normal operators.

Entire vectors for unbounded normal operators and abstract Cauchy problems

LET A be a closed operator on a Banach space X, and C~∞ (A)=∩_n~∞=1,D(A~n). The C~∞ vec-tor x is an entire vector for A if sum from n=0 to ∞ (t~n)/(n!)||A~n x|| 0. We will write e(A) for the set of all entire vectors for A. It is well known that the set of entire vectors for a self-adjoint operator is dense. Inthis note, we generalize this result to the situation of (unbounded) normal operators. We will

Unitary invariants of spectral measures with the CGS-property

A «CGS-property» for the spectral measures is introduced and the classical results of determining complete systems of unitary invariants for self-adjoint and bounded normal operators on separable Hilbert spaces are extended to the class of spectral measures with this property. As a consequence, the above mentioned results are extended to unbounded normal operators on separable Hilbert spaces. Moreover, three different kinds of multiplicity are defined and it is shown that for the measures with the «CGS-property» they all coincide. In the last section some analogues of the multiplicity functions defined by Stone [14] are related to the total multiplicity.

Compact perturbations of normal operators in a Krein space

We prove the existence of the spectral function of, in general, an unbounded normal operator in a Krein space that is the perturbation of a fundamentally reducible normal operator N by an operator of a special class that contain N-compact operators in it.

On some classes of unbounded operators

In Chap. 3, we introduce and begin the study of some fundamental classes of unbounded operators. The most important ones for this book are symmetric operators and self-adjoint operators. For a densely defined symmetric operator, the deficiency indices are defined and investigated, and the von Neumann formula about the domain of the adjoint is obtained. As a consequence, some self-adjointness criteria for symmetric operators are derived. Among others, we give a short proof of Naimark’s classical theorem, which states that each densely defined symmetric operator has a self-adjoint extension on a possibly larger Hilbert space. Further, we define classes of operators (sectorial operators, accretive operators, dissipative operators) that will appear later as operators associated with sectorial forms or as generators of contraction semigroups. The last section of this chapter contains a brief introduction to unbounded normal operators.

Cosine families of unbounded normal operators

Cosine families of unbounded normal operators

Support of the joint spectral measure

AbstractFor a doubly commuting n-tuple of unbounded normal operators in a Hilbert space the joint spectral measure can be constructed and its closed support described as the joint spectrum of the given n-tuple. The same is here shown for larger, possibly uncountable, families of operators.

Approximate solutions of problems involving normal operators

The spectral theory for unbounded normal operators is used to develop a systematic method of approximating functions of operators with other, more easily computable functions, leading to a priori error estimates in the operator norm. In particular, polynomial approximations are obtained for resolvents and semigroups in terms of inverses and resolvents, respectively.

Joint numerical ranges for unbounded normal operators

For bounded operators, the theory of the joint numerical range has been developed by various authors [1,2,3,4,5]. Especially, the properties of commuting normal n-tuples are discussed in detail. Our purpose here is to show that many results in the above references still hold in the case of unbounded normal operators (see Theorem 2.3, Corrollary 3.5, Theorem 4.1, Theorem 4.2). Besides, the operator algebras are closely related to the theory of joint spectrum and joint numerical ranges in the boundedcase (cf. [1,3]). How about unbounded operators? It seems that one must consider unbounded operator algebras. Some work has been done in this direction for the joint spectrum of unbounded normal operators [9]. In the last section of this paper, we provide some intimate relations between the joint numerical range and the unbounded operator algebras for unbounded normal operators.

Estimates and regularization for solutions of some ill-posed problems of elliptic and parabolic type

In this paper, we examine, in a systematic fashion, some ill-posed problems arising in the theory of heat conduction. In abstract terms, letH be a Hilbert space andA: D (A)⊂H→H be an unbounded normal operator, we consider the boundary value problemu(t)=Au(t), 0<t<∞,u(0)=u 0∈D(A), $$\mathop {\lim }\limits_{t \to 0} \left\| {u\left( t \right)} \right\| = 0$$ . The problem of recoveringu 0 whenu(T) is known for someT>0 is not well-posed. Suppose we are given approximationsx 1,x 2,…,x N tou(T 1),…,u(T N) with 0<T, <…<T N and positive weightsP i,i=1,…,n, $$\sum\limits_{i = 1}^N {P_i = 1} $$ such that $$Q_2 \left( {u_0 } \right) = \sum\limits_{i = 1}^N {P_i } \left\| {u\left( {T_i } \right) - x_i } \right\|^2 \leqslant \varepsilon ^2 $$ . If ‖u t(0)‖≤E for some a priori constantE, we construct a regularized solution ν(t) such that $$Q\left( {\nu \left( 0 \right)} \right) \leqslant \varepsilon ^2 $$ while $$\left\| {u\left( 0 \right) - \nu \left( 0 \right)} \right\| = 0\left( {ln \left( {E/\varepsilon } \right)} \right)^{ - 1} $$ and $$\left\| {u\left( t \right) - \nu \left( t \right)} \right\| = 0\left( {\varepsilon ^{\beta \left( t \right)} } \right)$$ where 0<β(t)<1 and the constant in the order symbol depends uponE. The function β(t) is larger thant/m whent<T k andk is the largest integer such that $$(\sum\limits_{k = 1}^N {P_i (T_i )} )< (\sum\limits_{k = 1}^N {P_i (T_i )} = m$$ , which β(t)=t/m on [T k, m] and β(t)=1 on [m, ∞). Similar results are obtained if the measurement is made in the maximum norm, i.e.,Q ∞(u 0)=max{‖u(T i)−x i‖, 1≤i≤N}.

A Joint spectral theorem for unbounded normal opertors

AbstractA joint spectral theorem for an n-tuple of doubly commuting unbounded normal operators in a Hilbert space is proved by using the techniques of GB*-algebras.

Convexity, boundedness, and almost periodicity for differential equations in Hillbert space

There are three kinds of results. First we extend and sharpen a convexity inequality of Agmon and Nirenberg for certain differential inequalities in Hilbert space. Next we characterize the bounded solutions of a differential equation in Hilbert space involving and arbitrary unbounded normal operator. Finally, we give a general sufficient condition for a bounded solution of a differential equation in Hilbert space to be almost periodic.

On when Invariant Subspaces of Unbounded Normal Operators are Reducing

On when Invariant Subspaces of Unbounded Normal Operators are Reducing

Unbounded Normal Operators on Banach Spaces

Unbounded Normal Operators on Banach Spaces

Unbounded normal operators on Banach spaces

Introduction. In this paper a theory of unbounded normal operators on complex Banach spaces is developed which generalizes many features of the classical theory of maximal normal operators on Hilbert spaces. In particular the operators studied are closely related to the unbounded scalar type spectral operators introduced by Bade [1] and to a generalization of these operators defined here. The definition of these normal operators is given in terms of generators of strongly continuous groups of isometries contained in a commutative V* algebra [6]. Many of the techniques used were suggested by and generalize results in [21, 5, and 19]. In this paper all Banach spaces (X, '3 etc.) are defined over the complex numbers C, and all operators are (not necessarily bounded, closed, or densely defined) linear transformations with domain and range contained in the same Banach space. An operator R is called self conjugate iff iR generates a strongly continuous group of isometries (cf. [19]). This definition generalizes both the notion of an unbounded self adjoint operator on Hilbert space and a concept introduced by I. Vidav [21] for elements in an arbitrary Banach algebra. A norm closed commutative subalgebra % of the algebra [N] of all everywhere defined bounded operators on X is called a commutative V* algebra if every element S E W can be written as R + iJ with R and J self conjugate elements of W [6]. In ?2, it is shown that the weak closure of such an algebra is again a commutative V* algebra. Also if X is weakly complete a weakly closed commutative V* algebra contains a self conjugate resolution of the identity for each of its elements. A commutative V* algebra with the same property is also constructed when =ID*. The numerical range of an operator on a Banach space is defined in a way resembling that of [17] but more easily applied to unbounded operators. Several criteria are given in ?3 for a densely defined operator with real numerical range to have a self conjugate closure. Self conjugate operators R and J are defined to commute iff the groups generated by iR and iJ commute in the ordinary sense (cf. [20, p. 647]). It is shown that this definition agrees with other ideas of

The spectral theorem for unbounded normal operators

self adjoint operator has a unique positive self adjoint square root A1/2; and an arbitrary self adjoint operator has a unique representation = A+ ~ A~ with A+ and A self adjoint and positive and the range of each contained in the null space of the other. Write ( = (T*Γ) 1/2 and, for complex λ and r ^ 0, let E(\, r) be the null space projection of (| T — XI — r/) +. For compact subsets K of the complex plane E(K)= Λ and for any Borel set M, E{M) = V {E(K): K compact and K £ M) . It is shown that E is the unique spectral measure such that T = \xE(dX) .