On the classical Volterra spectral equation

Abstract

We study the characterizations of the classical Volterra operator based on a simple relation between its singular numbers and the eigenvalues of its imaginary part on \( L^2[0,1] \).

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Introduction Let be a complex Hilbert space equipped with the inner product (⋅, ⋅) and the induced norm ‖ ⋅ ‖. Denote by ( ) the Banach algebra of bounded linear operators acting on with the operator norm ‖ ‖ = sup {‖ ‖ ∶ ∈ }, ∈ ( ). ‖ ‖=1 The integral operator on 2[0, 1] defined as 1 ( )( ) = ∫ 0 ( , ) ( ) with some kernel ( , ) ∈ 2([0, 1]2). Then is a Hilbert-Schmidt operator. Recall that for an operator the spectrum ( ) = { ∈ ℂ ∶ - is not invertible} is a nonempty compact subset of the complex plane. For any compact linear operator in a Hilbert space the singular numbers ( ) are the distances from to the set of all operators of rank less than or equal to - 1 for ⩾ 1. Their squares are the eigenvalues of the compact self-adjoint nonnegative operator ∗ counted according to their multiplicities. (see e.g. [1]) In particular, 1( ) = ‖ ‖. Denote by the Volterra operator © 2026 D. Tsedenbayar, B. Ser-Od, W. Qiqige This work is licensed under a Creative Commons “Attribution-NonCommercial 4.0 International” license. D. Tsedenbayar et al. On the classical Volterra spectral equation 275 and its adjoint ( )( ) = ∫ 0 ( ) on 2[0, 1]. 1 ( ∗ )( ) = ∫ ( ) The Halmos (see [2]) calculation yields (see [3]): ( ) = 2 (2 - 1) for all ⩾ 1, in particular ‖ ‖ = 2 . Note that ∞ ∑ 2 = 1 . 2 =0 If ∈ ( ) and { } are the eigenvalues of , then ( ) = ∑ . It is well known that is quasi-nilpotent, compact and even a Hilbert-Schmidt operator. Due to its excellent properties, many scholars have studied Volterra operator pencils on Hilbert spaces and formulated many interesting results (see e.g. [4-13]). The set = { ∈ 2[0, 1] ∶ ( ) = 0, 0 ⩽ ⩽ } is an invariant subspace for , for 0 ⩽ ⩽ 1 (see [14, 15]). Recall that Re = 1 ( + ∗) = 1 , 2 2 where P is the orthogonal projection onto the constant functions. It is easy to see that the nonzero eigenvalues of the imaginary part Im = 1 ( - ∗) 2 are ± 1 (2 - 1) for ∈ ℕ, each having multiplicity 1 (see [15]). Then we have the spectral equation 1 1 2 (Im ) = (( ∗ ) 2 ) ∪ (- (( ∗ ) 2 )), where denotes the spectrum (see [16]). Recall that for ∗ the orthonormal basis of 2[0, 1] consisting of eigenfunctions 2 ( ) = √2 cos ( 2 + 1 ) 276 Letters DCM&ACS. 2026, 34 (2), 274-278 corresponding to eigenvalues for ⩾ 0 (see [17]). = 4 (2 + 1)2 2 We obtain the characterizations of the Volterra operator on 2[0, 1]. The Results Theorem 3. Suppose that is a Hilbert-Schmidt operator on 2[0, 1] such that + ∗ = is a rank-one projection, quasinilpotent and 1 1 2 (Im ) = (( ∗ ) 2 ) ∪ (- (( ∗ ) 2 )). Then = . Proof. If is quasinilpotent and rank-one projection (3) and the spectra on both sides in (3) are countable set, then is a compact operator (see [18, 19]). On the other hand, we observe that 0 is not an eigenvalue of , and that and ∗ have no proper common invariant subspace. Also is rank-one projection and Im A ⩾ 0 then is unitary equivalent to the Volterra operator (see [20]). Suppose that the eigenvalues of Im A and the singular numbers { } ∈ℕ of have multiplicity 1. From (3), we have ∞ ∑ 2 = ( ∗ ) = (( 1 - Im A)( 1 2 + Im A)) = 2 =1 = 1 2 4 ( ) + 2 ([ , Im A]) + ((Im A) ) = ∞ = 1 + 2 ∑ ( 2 ) = 1 + 1 ∞ ∑ 2 . Hence, 4 =1 2 ∞ 4 2 =1 1 ∑ 2 = . 2 =0 Therefore = . This completes the proof. □ Conclusions We have seen that characterizations technique for the Volterra operator is based on some relation between its singular numbers and the eigenvalues of its imaginary part on 2[0, 1].