On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

and Applied Analysis 3 = 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 x − N


Introduction
Compact operators in infinite-dimensional separable Hilbert spaces are of relevance in the study of certain relevant applied problems in control theory and signal theory, [1].A natural property of such operators is that they can be represented with expansions using two orthogonal or orthonormal reciprocal bases of the separable Hilbert space.If the bases are orthonormal then both of them coincide so that this basis is autoreciprocal and then the formal study is facilitated [1,2].Many of the involved operators in mapping map an input space into an output space in the above problems are in addition self-adjoint.Another property of such operators is that they admit truncations using a finite number of the members of the orthonormal basis so that the truncated operators are also compact in a natural way, [1,2].The truncated operator describes a natural orthogonal projection of the involved vectors of the Hilbert space into a finitedimensional space whose dimension is deceased as the number of members of the basis used for its representation decreases.On the other hand, important attention is being devoted to many aspects of fixed point theory in metric, Banach, and more general spaces including the study of mappings being contractive, nonexpansive, asymptotically contractive, asymptotically nonexpansive, quasi-nonexpansive, Kannan and Meir-Keeler and cyclic-type contractions, and so forth.Also, it has been studied the relevance of the theory in properties in both general theory and applications such as the existence and uniqueness of solutions in differential, difference, and hybrid equations as well as in continuoustime, discrete-time, and hybrid dynamic systems, stability theory in the above problems [3][4][5][6][7], the existence/uniqueness of fixed points and best proximity points, and the boundedness of iterated sequences being constructed through the maps and the convergence of such iterated calculations to limit points.See, for instance, [3][4][5][6][8][9][10][11][12][13][14][15] and the references therein.The investigation of existence and uniqueness of common fixed points and best proximity points for several mappings and related properties is also important [10][11][12].The study of fixed and best proximity points has also inherent study of convergence of sequences to such points.Other studies of properties of convergence of sequences and operator sequences have been described in different problems as, for instance, the research on approximating operators and approximation theorems that of sigma convergence of double sequences or that of lamda-statistical convergence and summability.See, for instance, [13][14][15][16][17] and the references therein.
This paper is devoted to the investigation of self-adjoint compact operators in separable Hilbert spaces, their finitedimensional truncated counterparts, and the relations inbetween the corresponding properties for the norms of the 2 Abstract and Applied Analysis mutual errors end the errors in-between the corresponding fixed points and their respective convergence properties when iterated calculations through the operators are performed.Some examples of interest in signal theory and control theory are also given.The operators and the iterated sequences constructed through them are studied by using the expansions of the operators and their finite dimensional truncated versions by using a numerable orthonormal basis of the involved Hilbert space.

Preliminaries and Main Results
The following result includes some properties related to the approximations of  ∈  and  ∈  through orthonormal systems of different dimensions, complete orthonormal systems in , and orthonormal basis, that is, a maximal orthonormal system; that is, it is not a proper subset of any orthonormal system of , where  and  are an inner product space and a Hilbert space, respectively.Note that in the case where  is separable, a complete orthonormal system is always an orthonormal basis and vice versa.Lemma 1.Let  be an inner product space of inner product ⟨⋅, ⋅⟩ :  ×  → C (or R) endowed with a norm ‖ ⋅ ‖ :  → R 0+ defined by ‖‖ = ⟨, ⟩ 1/2 for any  ∈ , where R 0+ = { ∈ R :  ≥ 0}, let {  }  =1 and {  }  =1 be a finite orthonormal system in  and a given finite or numerable sequence of scalars, respectively, and let  and  be given integers fulfilling 1 ≤  ≤  ≤ ∞.If  = ∞ then {  }  =1 is, in addition, assumed to be numerable.Then, the following properties hold for any  ∈ .
Note that Property (vi) of Lemma 1 quantifies an approximation of an element of a finite-dimensional Hilbert space  via an orthonormal system in  of smaller dimension than that of such a space.Property (vii) relies on the approximation of an element in an infinite-dimensional separable Hilbert space by using a numerable orthonormal basis of .Lemma 2. Let  :  →  be a linear, closed, and compact self-adjoint operator in an infinite-dimensional separable Hilbert space  with a numerable orthonormal basis of generalized eigenvectors {  } ∞ =1  :  → .Then, the following properties hold:  (10) with    =  Ω  (  )() ⊕ (I −  Ω  (  ))() where  Ω  (  ) () ≡  Ω  (  ), for all ,  ∈ N, for all  ∈ .
Proof.Note that there is a numerable orthonormal basis for  since  is separable and infinite dimensional.Such a basis {  } ∞ =1 can be chosen as the set of generalized eigenvectors of the linear self-adjoint  :  →  since it is closed and compact and then bounded Also, since the linear operator  :  →  is closed and compact, the spectrum () of  :  →  is a proper nonempty (since  :  →  is infinite dimensional and bounded since it is compact) subset of C and numerable and it satisfies () =   () ∪ {0}, with   () ∪   () = {0}, where   (),   (), and   () are the punctual, continuous, and residual spectra of  :  → , respectively.Note that {0} ∈ () is also an accumulation point of the spectrum () since  is infinite dimensional and  :  →  is compact.Also, since  is separable, the spectrum of  :  →  is numerable, and ⟨  ,   ⟩ =   ; for all ,  ∈ N, one gets where   () = ⟨  ,   ⟩ is an eigenvalue of  :  → ; that is,   () ∈ (), associated with the eigenvector   since so that so that, except perhaps for reordering, |  ()| ≥ | +1 ()|, for all  ∈ N with {  ()} → 0 since  is separable and () is numerable.Assume that for any positive integer  the following identity is true: Then, since {  } ∞ =1 is an orthonormal basis of generalized eigenvectors, where   is the Kronecker delta.Then,    () = ⟨  ,   ⟩  ∈ (  ).Furthermore,   :  →  is compact as it follows by complete induction as follows.Assume that   :  →  is compact, then it is bounded.Note also that   :  →  is self-adjoint by construction and then normal.Thus,  +1 = (  ) :  →  is compact since it is a composite operator of a bounded operator   :  →  with a compact operator  :  → .Then, by complete induction,    () = ⟨  ,   ⟩  → 0 (∈ (  )) as  → ∞, for any  ∈ N since   :  →  is compact and  is infinite dimensional.Also, where P  is the projection operator of  on the onedimensional subspace   generated by the eigenvector   so that P   = ⟨,   ⟩  → 0 as  → ∞, for all  ∈ .Thus, Property (i) has been proved.To prove Property (ii), take an orthonormal basis associated with the set of finitedimensional eigenspaces of the respective eigenvalues.Note from Cauchy-Schwarz inequality that for some real constant  ∈ (0, 1), where {  } ∈N is a nondecreasing sequence of finite nonnegative integers defined by   = ∑ −1 =1   being built such that each   for  ∈ N accounts for the total of the dimensions   of the eigenspaces Ω  associated with the set of eigenvalues { 1 (),  2 (), . . .,  −1 ()} previous to   () for  ∈ N after eventual reordering by decreasing moduli.Then, lim  → ∞ |  ()|  = 0, for all  ∈ N, and where +  :  = 0, 1, . . .,  −1 } is now a set of   linearly independent elements belonging to the orthonormal basis of  that generate the eigenspace Ω  associated with   () with  (0) +  =  +  being an eigenvector and { () is a set of complex coefficients.Then,     → 0 as → ∞, for all  ∈ N from (20), so that lim  → ∞ (P  (  )) = {0}(∈   ).If there are some multiple eigenvalues, with all being of finite multiplicity since the operator  :  →  is compact, the above expression may be reformulated with projections on the finite-dimensional eigenspaces associated to each of the eventually repeated eigenvalues leading to lim  → ∞ (P Ω  (  )) = {0}(∈ Ω  ), for all  ∈ N. Note that and Property (ii) has been proved.Lemma 2 becomes modified for compact operators on a finite-dimensional Hilbert space as follows.(ii) If, in addition, ‖‖  ≤   for some real constants  ∈ (0, 1 ) and  ≥ 1, then Outline of Proof.First note that the spectrum of  :  →  is nonempty since the operator is self-adjoint.Note also that, since the Hilbert space is finite-dimensional Hilbert space, any set of normalized linearly independent eigenvectors of a self-adjoint operator is an orthonormal basis of such a Hilbert space [1].Property (i) is a direct counterpart of Property (i) of Lemma 2 except that {0} can be a value of the punctual spectrum of  :  →  but it is not an accumulation point of such a spectrum () since the Hilbert space is finitedimensional.Therefore, the result ⟨  ,   ⟩ → 0 as  → ∞ of Lemma 1 does not hold.Then, Property (i) follows directly from the above considerations.Property (ii) follows from the relations Remark 4. It turns out that Lemma 2 (ii) and Lemma 3 (ii) also hold if  :  →  is not self-adjoint since the corresponding mathematical proofs are obtained by using an orthonormal basis formed by all linearly independent vectors generating each of the subspaces.However, if the operator is not self-adjoint or if it is infinite dimensional while being selfadjoint, the set of (nongeneralized) eigenvectors is not always an orthogonal basis of the Hilbert space.
In the following, we relate the properties of operators on  with their degenerate versions obtained via truncations of their expanded expansions.Theorem 5. Let  be a separable Hilbert space and let () :  →  be a linear degenerated -finite-dimensional approximating operator of the linear closed and compact selfadjoint operator  :  → .Then, the following properties hold.
Note that Theorem 5 (ii) cannot be generalized, in the general case, for the case of a finite dimensional approximating linear operator () :  →  of smaller dimension  <  to any linear degenerated operator  :  →  of (finite) dimension .The reason is that the property that 0 ∈ () does not any longer hold, in general if  :  →  is finite dimensional.On the other hand, a way of describing the operator  :  →  and its approximating finitedimensional counterpart () :  →  is through the absolute error operator T(≡  − ()):  → .This is useful if either  :  →  is finite dimensional of dimension  >  where  is the dimension of () :  →  or if  :  →  is nondegenerated.Another useful characterization is the use of the relative error operator T() :  →  satisfying the operator identity () = (I + T()).Another alternative operator identity  = ()(I + T1 ()) cannot be used properly if  :  →  is infinite dimensional since () :  →  is degenerated of finite dimension .We discuss some properties of the operator identity () = (I + T()) through the subsequent result.Lemma 6.Let  be a separable Hilbert space and let  :  →  be a nonnull and nondegenerated (i.e., of infinitedimensional image) linear closed and compact operator and let () :  →  be the linear degenerated -finite-dimensional approximating operator of  :  → .Then, there is an operator T() :  →  such that () can be represented by () = (I + T()), Dom( T()) ⊆ Dom(), and Im( T()) ⊆ Dom() with the following properties.(ii) The operator  T() :  →  is nondegenerated, unique, and compact.
(iii) The minimum modulus of  :  →  is ( T()) = 0 so that if it is invertible, its inverse is not bounded.If  :  →  is degenerated, that is, finite dimensional of dimension  > , injective with closed image then its minimum modulus is positive and finite.If, furthermore,  :  →  is invertible then T() :  →  is a compact operator with bounded minimum modulus ( T()).
Proof.The existence of such an operator T() :  →  is proved by construction.Let {  } ∈N be an orthonormal basis of generalized eigenvectors of  :  →  and {V  } ∈N an orthonormal basis of T() :  → , respectively.Then, one gets for some sequences of complex coefficients {  } ∈N , for all  ∈ N, (44) Equations ( 43) are also satisfied with   = 0, for all  ∈ , for all  ∈ N, and all  > .Thus, T :  →  is then non-unique, in general.Properties (i)-(ii) have been proved.Now, let (Γ) = {inf ‖Γ‖ :  ∈ , ‖‖ = 1} be the minimum modulus of the linear operator Γ :  → .If ‖‖ = 1, then if  :  →  is injective with closed image (this implies that such an image is finite dimensional), then () > 0 and since ,  :  →  are both bounded since they are compact, one gets If  :  →  is infinite dimensional, then () = 0 and it cannot then have bounded inverse.If  :  →  is degenerated of dimension  = , then T() is the null operator with ( T) = 0.If  :  →  is degenerated of dimension  >  and invertible, then :  →  is bounded and compact since it is a composite operator of a compact operator ( − ()) on  and a bounded operator  −1 on .Property (iii) has been proved.

Examples
Hilbert spaces for the formulation of equilibrium points, stability, controllability [16,18,19], boundedness, and square integrability (or summability in the discrete formalism) of the solution in the framework of square-integrable (or squaresummable) control and output functions are of relevant importance in signal processing and control theory and in general formulations of dynamic systems, in general.See, for instance, [1,2,7,9,16,17,19,20] and the references therein.Two examples with the use of the above formalism to dynamic systems and control issues are now discussed in detail.