By Joachim Weidmann (auth.)

This English variation is nearly similar to the German unique Lineare Operatoren in Hilbertriiumen, released by way of B. G. Teubner, Stuttgart in 1976. a number of proofs were simplified, a few extra workouts were integrated, and a small variety of new effects has been extra (e.g., Theorem 11.11 and Theorem 11.23). moreover quite a few minor error has been corrected. Frankfurt, January 1980 J. Weidmann vii Preface to the German variation the aim of this ebook is to offer an advent to the speculation of linear operators on Hilbert areas after which to continue to the attention-grabbing applica tions of differential operators to mathematical physics. along with the standard introductory classes universal to either mathematicians and physicists, just a primary wisdom of advanced research and of normal differential equations is believed. crucial result of Lebesgue integration concept, to the level that they're utilized in this publication, are compiled with entire proofs in Appendix A. i'm hoping for this reason that scholars from the fourth semester on can be in a position to learn this publication with no significant hassle. although, it will probably even be of a few curiosity and use to the instructing and learn mathematician or physicist, due to the fact between different issues it makes simply available numerous new result of the spectral concept of differential operators.

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**Example text**

In Hilbert spaces it is useful to introduce another notion of dimension. The dimension (more precisely, the Hubert space dimension) of a Hubert space H is the cardinality of an ONB of H. 10(c) this dimension does not depend on the choice of the ONB. 9(d) for finite dimensional Hilbert spaces the two definitions of dimension coincide; for infinite dimensional spaces this is not the case, cf. 8. Proposition. There exist Hubert spaces of arbitrary (Hubert space) dimension. PROOF. Let be an arbitrary cardinal number, and let A be a set of with = I for all cardinality K.

For the subspace T2, defined uniquely by part (c) of this theorem, we write briefly T2 = T T1. The subspace T2 is called the orthogonal complement e of T1 with respect to T. For T = H we T1 fE obtain that H e T1 = is closed (the proof for T2 goes the same way). Let be a sequence from T1 such that and let T2. Hence f E T1 T2= T1 we have c Since T2 and thus we have f= g1 + g2 with g1 E T1, g2 E T2. On the other hand, it follows from f,, E T1 that and so fI T2, and, consequently, g2 =f— g1 E T2 n Ti-.

EXERCISES a Hubert space and let A be a countable subset of H such that L(A) = H. 8. Let H be dimension exists. Hint: Apply the Schmidt orthogonalization process to A; for the resulting ONB M we have L(M) = H. (It can actually be proved that no infinite dimensional Hubert space can have an algebraic dimension smaller than the cardinality of the continuum; cf. N. Bourbaki [2], Chap. 9. >) be a pre-Hilbert space. For any n elementsf1,.. of H the Gram determinant is defined by D(f1,. , = det (