91Ó°ÊÓ

The adjoint of a linear operator is a new operator that is defined in terms of the inner product. For the Volterra operator \(V\), its adjoint \(V^*\) is defined as: \[ \langle Vf, g \rangle = \langle f, V^*g \rangle, \] where \(f\) and \(g\) are continuous functions, and \(\langle \cdot, \cdot \rangle\) denotes the inner product. To compute the adjoint of the Volterra operator, consider the left-hand side of the above equation: \[ \langle Vf, g \rangle = \int_0^1 (Vf)(x)g(x)dx = \int_0^1 \left( \int_0^x f(t)dt \right) g(x)dx. \] Now, we will use integration by parts to rewrite this expression. Let \(u = g(x)\) and \(dv = \int_0^x f(t)dt\). Then, we have: \[ du = g'(x)dx, \;\;\; v = f(x) \] By integrating by parts, we obtain: \[ \int_0^1 \left( \int_0^x f(t)dt \right) g(x)dx = \left[ f(x)g(x) \right]_0^1 - \int_0^1 f(x)g'(x)dx \] Since \(f\) and \(g\) are continuous functions, this expression simplifies to: \[ \langle Vf, g \rangle = f(1)g(1) - \langle f, g' \rangle \] Comparing this expression to the definition of the adjoint, we see that: \[ V^*g = g(1) - g'. \] So, the adjoint of the Volterra operator is given by \(V^*g = g(1) - g'\).

Now we need to find the product of the Volterra operator and its adjoint, \(VV^*\). The action of this operator on a function \(g\) can be expressed as: \[ (VV^*g)(x) = (V(V^*g))(x) = \int_0^x (V^*g)(t)dt. \] Using the expression for the adjoint we derived in Step 1 (i.e., \(V^*g = g(1) - g'\)), we get: \[ (VV^*g)(x) = \int_0^x (g(1) - g'(t))dt. \]

To find the eigenvalues of the operator \(VV^*\), we need to solve the following eigenvalue problem: \[ (VV^*g)(x) = \lambda g(x), \] where \(\lambda\) is an eigenvalue and \(g\) is the corresponding eigenfunction. Using the expression for \(VV^*g(x)\) from Step 2, we can write the eigenvalue problem as: \[ \int_0^x (g(1) - g'(t))dt = \lambda g(x). \] This is an important equation, but due to the complexity of the problem, finding its eigenvalues and eigenfunctions is not possible in a closed-form. Therefore, singular values cannot be obtained explicitly in general. However, they can be approximated numerically, revealing some insights into the properties of the singular value spectrum.

To find the singular values of the Volterra operator, we need to take the square root of the eigenvalues (\(\lambda\)) found in Step 3: \[ \sigma = \sqrt{\lambda}. \] As mentioned earlier, the eigenvalues and singular values cannot be expressed in a closed form in general. However, their approximate numerical values can be computed and analyzed, allowing the study of their behavior as the input function varies.