169-174 Mahendra Sahi R C O S T N M. Sahi / BIBECHANA 11(1) (2014) 169-174: (Online Publication: March, 2014) p.169 BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Some special characterisations of Fredholm operators in Banach space Mahendra Shahi Department. of Mathematics, M.M.A.M. Campus, Biratnagar Tribhuvan University, Nepal E-mail: mshahi11@hotmail.com Accepted for publication: February 06, 2014 Abstract A bounded linear operator which has a finite index and which is defined on a Banach space is often referred to in the literature as a Fredholm operator. Fredholm operators are important for a variety of reasons, one being the role that their index plays in global analysis. The aim of this paper is to prove the spectral theorem for compact operators in refined form and to describe some properties of the essential spectrum of general bounded operators by the use of the theorem of Fredholm operators. For this, we have analysed the Fredholm operator which is defined in a Banach space for some special characterisations. © 2014 RCOST: All rights reserved. Keywords: Bounded linear operator; Compact operator; Fredholm operator; Banach space. 1. Introduction An operator K defined by a kernel k is called a Fredholm type operator. The name goes back to Swede, E. Ivar Fredholm who developed a comprehensive theory for integral equations of second kind at the beginning of the twentieth century [4]. Let X = Y = C[a,b] be a Banach Space.Let k(s,t) be defined for a≤s≤band a≤t≤b. Then for each xϵX the Riemann integral � k�s, t�x�t�dt �1� � exists and defines a continuous function of s on [a,b]. The integral (1) defines a linear operator K on X into X. If we take Kx = y to mean y�s� = � k�s, t�x�t�dt �2� � the equation(2) is known as Fredholm type integral equation of the first kind [7]. Another operator T is obtained by defining Tx = y to mean M. Sahi / BIBECHANA 11(1) (2014) 169-174: (Online Publication: March, 2014) p.170 y�s� = x�s� − � k�s, t�x�t�dt �3� � Here k(s,t) is a continuous function on [a,b]×[a,b] and is called the kernel of the integral equation. y(s) is continuous on [a,b] and therefore yϵC[a,b]. The equation(3) is known as Fredholm type integral equation of the second kind [5]. Equations of this sort are of great importance. The application of this operator plays a vital role in the theory of boundary value problems in differential equations. Fredholm studied Fredholm type integral equations of the second kind, which gave rise to such operators. In view of the development of the theory of Fredholm operators the following definitions are frequently used. Definitions Fredholm operator (i) A closed linear operator which has a finite index is called a Fredholm operator. (ii) Let X and Y are Banach spaces. A linear operator T from X to Y is called a Fredholmoperator if i. T is closed. ii. The domain of T is dense in X. iii. α(T), the dimension of the null space N(T) of T is finite. iv. The range of T is closed in y v. B(T), The co-dimension of R(T) in Y is finite. The terminology stems from the classical theory of integral equations. Special types of Fredholm operators were considered by many authors since that time but systematic treatment were not given until the work of Atkimon [1], Gohberg [4] and yood [9]. A general account of the history of the theory is given by [a]Gohberg Krein [3] and (b) Kato [5]. For a good general account of the theory can be found in the book written by Gohberg [4]. Definition(3) Let B & C be two Banach spaces. A bounded linear operator A: B → C is defined to be a linear map for which the norm ∥ A ∥∶= sup�∥ Af ∥∶ ∥ f ∥≤ 1 � is finite. Definition(4) Let X and y be normed linear spaces. Suppose T is a linear operator with domain X and range in y. We say that T is compact if for each bounded sequence {xn} in X, the sequence {T xn} contains a sub sequence converging to some limit in y. A compact operator is also called completely continuous. Lemma (1) If A is a compact operator on B, then (λI- A) is Fredholm for all λ ≠ 0. Proof: We first prove that ℒ≔Ker(λI - A) is finite dimensional by contradiction. If this were not the case there would exist an infinite sequence xnЄℒ such that ∥ x� ∥ = 1 and ∥ x − x� ∥ ≥ 1/2 for all distinct m and n. Since Axn = λxn and λ ≠ 0, we could conclude that Axn has no convergent subsequence. We can write B=ℒ + M, where ℒ ∩ M = {0}and M is a closed linear subspace on which (λI - A) is one-one. We next prove that ℛ≔Ran(λI-A) is closed. If gnЄ ℛand ∥ #$ − # ∥ →0, then there exist fn Є M such that gn= (λI- A)fn. If ∥ f� ∥ is not a bounded sequence then by passing to a subsequence (without M. Sahi / BIBECHANA 11(1) (2014) 169-174: (Online Publication: March, 2014) p.171 change of notation) we may assume that ∥ f� ∥ → ∞ as n → ∞. Putting hn≔ fn/∥ %$ ∥we have∥ ℎ$ ∥ =1 and kn≔ (λI - A)hn → 0. The compactness of A implies that h n = λ-1(Ahn + kn) has a convergent subsequence. Passing to this subsequence we have hn → hwhere ∥ ℎ ∥ = 1, h Є M, andh = λ-1Ah. We conclude that h Є M ∩ ℒ. The contradiction implies that ∥ %$ ∥ is a bounded sequence. Given this fact the compactness of A implies that the sequence fn = λ-1(Afn + gn) has a convergent subsequence. Passing to this subsequence we obtain fn→fas n → ∞, sof = λ-1(Af + g), and g = (λI - A)f. Therefore ℛ is closed. Since Ran(λI - A) is closed, an application of the Hahn-Banach theorem implies that its codimension equals the dimension of Ker(λI - A*) in B*. But A* is compact, so this is finite by the first paragraph. Our next theorem provides a second characterization of Fredholm operators. Theorem (1) Every Fredholm operator has closed range. The bounded operator A : B → C is Fredholm if and only if there is a bounded operator B : C → B such that both (AB - I) and (BA - I) are compact. Proof: If A is Fredholm then B1≔ Ker(A) is finite-dimensional and so has a complementary closed subspace B0 in B. Moreover A maps B0one-one onto Co ≔ Ran(A). If C1 is a complementary finite-dimensional subspace of C0 in C then the operatorX : B0⨁ C1 → C defined by (�% ⨁ )� ≔ *% + ) is bounded and invertible. We deduce by the inverse mapping theorem that C0≔X(B0) is closed. This completes the proof of the first statement of the theorem. Still assuming that A is Fredholm, put B(g⨁v) ≔(A 0) -1g for all g Є C0 and v Є C1, where A0 : B0 → C0is the restriction of A to B0. Then Bis a bounded operator from C to Band both of ,- ≔ *. − /, ,0 ≔ .* − / �1� are finite rank and hence compact. Conversely suppose that A, Bare bounded, K1, K2are compact and (1) hold. Then ,12�*� ⊆ ,12�/ + ,0�, 456�*� ⊇ 456�/ + ,-�. Since (I + K1) and (I + K2) are both Fredholm by Lemma (7.1), it follows that Amust be Fredholm. The proof of Theorem (1) provides an important structure theorem for Fredholm operators. Theorem (2) If A is a Fredholm operator then there exist decompositions B = B0⨁ B1 and C = C0⨁ C1 such that (i) B0and C0 are closed subspaces; M. Sahi / BIBECHANA 11(1) (2014) 169-174: (Online Publication: March, 2014) p.172 (ii) B1and C1 are finite-dimensional subspaces; (iii) B1 = Ker(A) and C0= Ran(A); (iv) Index(A) = dim(B1) - dim(C1); (v) A has the matrix representation * = 9*: 0 0 0< �2� where A0 : B0 → C0 is one-one onto. Example (1) Let A be a Fredholm operator on the Banach space B. Prove that if Ker(A) = {0} then Ker(A - I) = {0} for all small enough . It can be proved that if Ran(A) = Bthen Ran(A - I) = B for all small enough . Before stating our next theorem we make some definitions. We say that λ lies in the essential spectrum EssSpec(A) of a bounded operator A if (λI - A) is not a Fredholm operator. Since the set κ (B) of all compact operators is a norm closed two-sided ideal in the Banach algebra ℒ(B) of all bounded operators on B, the quotient algebra C := ℒ(B)/ κ(B) is a Banach algebra with respect to the quotient norm ∥ >�*� ∥≔ inf �∥ * + , ∥∶ , ∈ κ�.�� where π : ℒ(B) → Cis the quotient map. The Calkin algebra C enables us to rewrite Theorem (1) is particularly in a simple form. Theorem (3) The bounded operator A on B is Fredholm if and only if π(A) is invertible in the Calkin algebra C. If A Є ℒ(B) then BCCDE1F�*� = DE1F�>�*��. Proof : Both statements of the theorem are elementary consequences of Theorem (1). Corollary: If A : B → B is a Fredholm operator and B ≔ A + K where K is compact, then B is a Fredholm operator and BCCDE1F�*� = BCCDE1F�.�. Theorem (4) If A is a Fredholm operator on B then A* is Fredholm. Proof : Suppose that AB = I + K1 and BA = I + K2 where K1, K2 are compact. Then B*A* = I + K1* and A*B* = I + K2*. We deduce that A* is Fredholm by applying Theorem (1). M. Sahi / BIBECHANA 11(1) (2014) 169-174: (Online Publication: March, 2014) p.173 Example (2): Prove directly from the definition that if A1 and A2 are both Fredholm operators then so is A1A2. Note: If B1 = B2 then this is an obvious consequence of Theorem (7.3), but there is an elementary direct proof. Theorem (5) If A:B → C is a Fredholm operator, then there exists Є >0 such that every bounded operator X satisfying ∥ ( − * ∥<∈ is also Fredholm with H6I1J�(� = H6I1J�*�. Proof: We make use of the matrix representation of Theorem (2). If ( = 9. K L B< and ∥ ( − * ∥<Є then ∥ . − *: ∥< cϵ, so B is invertible provided ϵ > 0 is small enough. If f ϵ BOand g ϵ B1then X(f ⨁g) = 0 if and only if .% + K# = 0, L% + B# = 0. This reduces to �B – L.N-K�# = 0, where (E – DB-1 C)g : B1 → C1, both of these spaces being finite-dimensional. We deduce that IHO�,12�(�� = IHO�,12�B – L.N-K�� for all small enough Є > 0. By applying a similar argument to (∗ = 9.∗ L∗ K∗ B∗< we obtain IHO�KQR12�(�� = IHO�KQR12�B − L.N-K�� for all small enough ε > 0. Problem now implies that H6I1J�(� = H6I1J�B − L.N-K�� = IHO�ℬ-� − IHO�K-�. This formula establishes that index(X) does not depend on X, provided ∥ ( − * ∥is small enough. Theorem (5) establishes that the index is a homotopy invariant: if t → A t is a norm continuous family of Fredholm operators then index(At) does not depend on t. In a Hilbert space context one can even identify the homotopy classes. M. Sahi / BIBECHANA 11(1) (2014) 169-174: (Online Publication: March, 2014) p.174 Conclusion Thus, we see that Fredholm operators are important for variety of reasons, one being the role that their index plays in global analysis. The dimension of null space N(T) and the co-dimension of R(T) of the operator T are finite and it is closed and range of the operator is also closed. The application of this operator plays a vital role in the theory of boundary value problems in differential equations. References [1] F-V Atkinson, Acta Sci. Math., 15 (1953) 38. [2] Bovenbek Boss, J. Phillips, Canada. J. Math., 57 (2005) 225. [3] I.C. Gohberb, M. G. Krein, Amer. Math. Soc. Transl., 13 (1960)185. [4] S. Goldberg, Unbounded linear operators, Mc Graw Hill, New York, 1966. [5] T Kato, Perturbation theory for linear operators, Springer, 1996. [6] G. Kothe, Topological Vector Spaces, Vol. 1.and Vol. II, 1979. [7] A.E. Taylor, D. C. Lay, Introduction Functional Analysis, 2nd Ed. Wiley, New York, 1980. [8] A.E. Taylor, Theorems on ascent, descent, nullity and defect of linear operators, Math. Ann., 163 (1966) 18. [9] B. Yood, Properties of linear transformations: Preserved under addition of a completely continuous trans. Duke Math., J. 18.