We investigate the spectrum of a differential operator of fourth order with bounded operator coefficients and find a formula for the trace of this operator. | Turk J Math 28 (2004) , 231 – 254. ¨ ITAK ˙ c TUB The Trace Formula for a Differential Operator of Fourth Order With Bounded Operator Coefficients and Two Terms Erdal G¨ ul Abstract We investigate the spectrum of a differential operator of fourth order with bounded operator coefficients and find a formula for the trace of this operator. Key Words: Hilbert Space, Self-adjoint operator, Kernel operator, Spectrum, Essential spectrum, Resolvent. 1. Introduction Let H be a separable Hilbert space of infinite dimension. Consider the operators L0 and L in the space H1 = L2 (H; []) which are formed by differential expressions l0 (y) = yıv (x), l(y) = yıv (x) + Q(x)y(x) with the same boundary conditions y0 (o) = y0 (π) = 0 and y000 (0) = y000 (π) = 0, respectively. Suppose that the operator function Q(x) in the expression l(y) satisfies the following conditions: 1. For every x ∈ [0, π], Q(x) : H → H is a self adjoint kernel operator. Moreover, Q(x) has weak derivative of second order in this interval and for x ∈ [0, π], Q(i) (x) : H → H are self-adjoint operators (i = 1, 2). 2. ||Q||H1 ||Q||H1 (m = 0, 1, 2, .). (5) For the self adjoint operator R0λ = (L0 − λI)−1 , since || R0λ ||H1 = max |λ − m4 |−1 , then m from (5) we can write ||R0λ ||H1 < ||Q||−1 H1 . Because of this, we have ||QR0λ||H1 ≤ ||Q||H1 · ||R0λ||H1 < 1. By considering this inequality, we conclude that A(B) = R0λ − BQR0λ is a contraction operator from L(H1 , H1 ) to L(H1 , H1 ), where B ∈ L(H1 , H1). In this case, it is known that there exists an unique solution B = B0 which belongs to the space L(H1 , H1 ) of the equation R0λ − BQR0λ = B. Moreover, since R0λ − Rλ QR0λ = Rλ we have Rλ = B0 ∈ L(H1 , H1 ) and so λ ∈ ρ(L) (resolvent set of L). Hence, the spectrum of L is a subset of the union of the pairwise disjoint intervals m4 − ||Q||H1 , m4 + ||Q||H1 , 234 ¨ GUL 4 4 (m = 0, 1, 2, .) , . σ(L) ⊂ ∪∞ m=0 [m − ||Q||H1 , m + ||Q||H1 ]. From Lemma 1 and the equation Rλ = R0λ − Rλ QR0λ ,
