By Charles W. Groetsch
Spectral idea of bounded linear operators groups up with von Neumann’s idea of unbounded operators during this monograph to supply a common framework for the examine of strong equipment for the assessment of unbounded operators. An introductory bankruptcy presents quite a few illustrations of unbounded linear operators that come up in quite a few inverse difficulties of mathematical physics. sooner than the overall thought of stabilization equipment is built, an in depth exposition of the required historical past fabric from the speculation of operators on Hilbert house is equipped. numerous particular stabilization tools are studied intimately, with specific realization to the Tikhonov-Morozov procedure and its iterated version.
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Additional resources for Stable Approximate Evaluation of Unbounded Operators
Example text
This means that if {xn } ⊂ D(L), xn → x ∈ H1 , and Lxn → y ∈ H2 , then 32 2 Hilbert Space Background (x, y) ∈ G(L), that is, x ∈ D(L) and Lx = y. For example, the differentiation operator defined in the previous paragraph is closed. For suppose {fn } ⊆ D(L) and fn → f and fn → g, in each case the convergence being in the L2 [0, 1] norm. Since x fn (x) = fn (0) + fn (t)dt 0 we see that the sequence of constant functions {fn (0)} converges in L2 [0, 1] and hence the numerical sequence {fn (0)} converges to some real number C.
0 In this case the Tikhonov-Morozov approximation to g is found to be gαδ = L(I + αL∗ L)−1 f δ = π 2 ∞ π f δ (s) sin nsds αn sin nx n=1 0 where αn = n2 1 − e−n2 / 1+α n2 1 − e−n2 2 .
11. If L is closed and densely defined, then LL∗ is closed. Proof. Suppose yn ∈ D(LL∗ ), yn → y and LL∗ yn → u ∈ H2 . Then, (I +LL∗ )yn → y +u and hence, since L is bounded, yn → L(y +u). Therefore, L(y + u) = y, that is, y ∈ D(LL∗ ) and y + LL∗ y = y + u, or LL∗ y = u and hence LL∗ is closed. , f ∈ L2 [0, 1]}. , y ∈ L2 [0, 1], y(0) = y(1) = 0} and L∗ y = −y . , f ∈ L2 [0, 1], f (0) = f (1) = 0} = {f ∈ D(L) : f and L∗ Lf = −f . Now, given h ∈ L2 [0, 1], Lh = f if and only if f ∈ D(L∗ L) and h = (I + L∗ L)f .