Spectral Theory For Strongly Continuous Cosine

: Let ( C ( t )) t ∈ R be a strongly continuous cosine family and A be its infinitesimal generator. In this work, we prove that, if C ( t )−cosh λt is semi-Fredholm (resp. semi-Browder, Drazin inversible, left essentially Drazin and right essentially Drazin invertible) operator and λt ∈ ̸ iπ Z , then A − λ 2 is also. We show by counter-example that the converse is false in


Introduction
Let X be a complex Banach space, B(X) denote the algebra of all bounded linear operators on X and C(X) the set of all linear closed operators from X to X. We write D

(T), R(T), N(T), ρ(T), σ(T), σp(T), σap(T)
and σr(T) respectively for the domain, the range, the kernel, the resolvent, the spectrum, the point spectrum, the approximate point spectrum and residual spectrum of an operator T ∈ C(X). The function resolvent of T ∈ C(X) is de ned for all λ ∈ ρ(T) by R(λ, T) = (λ − T) − . The ascent a(T), the descent d(T), the essential ascent ae(T) and the essential descent de(T) of an operator T ∈ C(X) are de ned respectively by a(T) = inf{k ∈ N : N(T k ) = N(T k+ )}, d(T) = inf{k ∈ N : R(T k ) = R(T k+ )}, ae(T) = min{k ∈ N : dim N(T k+ )/ N(T k ) < ∞}, de(T) = min{k ∈ N : dim R(T k )/ R(T k+ ) < ∞}, with the convention inf(∅) = ∞, see ( [5,13]). For T ∈ C(X), if there is an operator S ∈ B(X) with R(S) ⊆ D(T) such that STS = S, TSx = STx for all x ∈ D(T), and T k (I − TS) = for some k ∈ N, then S is called a Drazin inverse of T. Note that an operator T ∈ C(X) has a Drazin inverse if and only if there exists k ∈ N such that a(T) = d(T) = k and X = R(T k ) ⊕ N(T k ), (see [12]). An operator T ∈ C(X) is a left essentially Drazin invertible operator if ae(T) < ∞ and R(T ae(T)+ ) is closed. If de(T) < ∞ and R(T ae(T) ) is closed, then T is called right essentially Drazin invertible. The Drazin invesible, left essentially Drazin and right essentially Drazin invertible spectra are de ned by An operator T ∈ C(X) is called upper semi-Fredholm (resp. lower semi-Fredholm) if the range R(T) is closed and dim N(T) < ∞ (resp. codimR(T) < ∞). If T is either upper or lower semi-Fredholm, then T is called a semi-Fredholm operator. If T is both upper and lower semi-Fredholm, then T is called a Fredholm operator, see ( [11]). The upper semi-Fredholm spectrum σ uf (T), the lower semi-Fredholm spectrum σ lf (T), the spectrum semi-Fredholm σ sf (T) and the Fredholm spectrum σ f (T) of T are de ned by We say that an operator T ∈ C(X) is upper semi-Browder if it is upper semi-Fredholm and has nite ascent. Similarly, T is lower semi-Browder if it is lower semi-Fredholm and has nite descent. An operator T is Browder if it is both lower and upper semi-Browder, see ( [13]). The upper semi-Browder spectrum σ ub (T), the lower semi-Browder spectrum σ lb (T), the spectrum semi-Browder σ sb (T) and the Browder spectrum Consider in X the well-posed Cauchy problem Where A : X −→ X is a densely de ned closed operator with nonempty resolvent set ρ(A). The problem (*) is (see [4,10]) well-posed if and only if A generates a strongly continuous cosine operator function (C(t)) t∈R , i.e., a family of operators satisfying the following conditions: 2. C( ) = I (the identity operator). 3. C(t)x is strongly continuous with respect to t for any xed x ∈ X.
There exist some M ≥ , ω ∈ R such that C(t) ≤ Me ωt for all t ≥ . If (C(t)) t∈R is a strongly continuous cosine operator function, then the in nitesimal generating operator A is de ned by A solution of problem (*) is given with the help of a strongly continuous cosine operator function by the formula u(t) = C(t)u + S(t)u for t ∈ R, where S(t) is the sine operator function associated with the (C(t)) t∈R and is de ned as S(t)x := t C(s)xds, t ∈ R, x ∈ X. In this work we will use the theory of integration in the sense of Bochner.
If (C(t)) t∈R is a uniformly continuous operator cosine function then there is an A ∈ B(X) with For t ∈ R, the function f : z ∈ C → cosh t √ z de nes an entire function. Thus, according to the spectral mapping theorem, we have cosh t σ * (A) = σ * (C(t)), for all t ∈ R, with σ * the spectrum corresponding to regularity in the sense of V. Müller [7, De nition 6.1].
In the context of a strongly continuous cosine the following spectral inclusion cosh t σ(A) ⊆ σ(C(t)), t ∈ R was obtained by B. Nagy; he also gave an example where the reverse inclusion fails [8] and he showed that σ * (C(t)) = cosh t σ * (A), t ∈ R, with * ∈ {p, r}. However there are several large classes of generators A for which the spectrum of C(t) can be expressed in terms of σ(A), namely σ(C(t)) = cosh t σ(A), t ∈ R, if A is the generator of a uniformly bounded cosine function on a Hilbert space [14] or of a cosine function of normal operators [15].
In this paper, we continue to study the spectral theory of strongly continuous cosine operator function. We investigate the relationships between the di erent spectra of a strongly continuous cosine operator function and their generators, precisely we prouve that where σ * denotes the upper and lower semi-Fredholm, semi-Fredholm, Fredholm, Drazin, upper and lower semi-Browder, semi-Browder, Browder, essential ascent and descent spectra. We show by counter-example that these inclusions are strict in general.

Main results
The following lemmas are among the most widely used results of this paper.
Then S λ (t) ∈ B(X) is an operator that commutes with A, and for all x ∈ X.
Proof. By Lemma 2.2, there exist two operators L λ (t), G λ (t) ∈ B(X) such that For all n ≥ and x ∈ X, we have L n λ (t)x ∈ D(A n ). In fact, the proof is by induction. For n = , from lemma 2.2

Lemma 2.4. Let A be the generator of a cosine operator function (C(t)) t∈R .
Then for all q ∈ N, t ≠ and λ ∈ C with λt ∉ iπZ, if R C(t) − cosh λtI q is closed, then R(A − λ ) q is also closed.
Proof. Let (yn) n∈N be a sequence in R(A − λ ) q converging to y ∈ X. Then there is a sequence (xn) n∈N of D(A q ) satisfying (A − λ ) q xn = yn . By Lemma 2.2, we obtain (A − λ ) q F λ,q (t)yn + H λ,q (t)S q λ (t)yn = yn .
Hence, we conclude that Thus, yn − (A − λ ) q F λ,q (t)yn ∈ R(C(t) − cosh λt) q . Since R(C(t) − cosh λt) q is closed and (A − λ ) q F λ,q (t) is bounded linear operator, it follows that the sequence yn − (A − λ ) q F λ,q (t)yn converges to y − (A − λ ) q F λ,q (t)y as n tends to ∞ and We obtain y ∈ R(A − λ ) q , which completes the proof.
Theorem 2.1. Let A be the generator of a cosine operator function (C(t)) t∈R . Then for all t ≠ , is a subspace of X of nite codimension. Therefore A − λ is lower semi-Fredholm. 3. It is easy by the previous assertions of this theorem. 4. Obvious.

Theorem 2.2. Let (C(t)) t∈R be a strongly continuous cosine function of operators with in nitesimal generator A. Then for all t ≠
We need the following lemma, which will also be useful later.