Minimax property

of eigen values

A special type of relationship connecting the eigen values of a completely-continuous self-adjoint operator $ A $( cf. also Completely-continuous operator) with the maximum and minimum values of the associated quadratic form $ ( A x , x ) $. Let $ A $ be a completely-continuous self-adjoint operator on a Hilbert space $ H $. The spectrum of $ A $ consists of a finite or countable set of real eigen values $ \lambda _ {n} $ having unique limit point zero. The root subspaces corresponding to the non-zero eigen values consist of eigen vectors and are finite dimensional; the eigen subspaces associated with distinct eigen values are mutually orthogonal; $ A $ has a complete system of eigen vectors. The spectral decomposition of $ A $( cf. Spectral decomposition of a linear operator) has the form: $ A = \sum \lambda _ {i} P _ {i} $, where $ \lambda _ {i} $ are the distinct eigen values, $ P _ {i} $ are the projection operators onto the corresponding eigen spaces, and the series converges in the operator norm. The norm of $ A $ coincides with the maximum modulus of the eigen values and with $ \max \{ {| ( A x , x ) | } : {x \in H, | x | = 1 } \} $; the maximum is attained at the corresponding eigen vector.

Let $ \lambda _ {1} ^ {+} \geq \lambda _ {2} ^ {+} \geq \dots $ be the positive eigen values of $ A $, where each eigen value is repeated as often as its multiplicity. Then

$$ \tag{1 } \left . \begin{array}{c} \lambda _ {1} ^ {+} = \ \max _ {x \in H } \frac{( A x , x ) }{| x | ^ {2} } , \\ \lambda _ {n+} 1 ^ {+} = \ \min _ {y _ {1} \dots y _ {n} } \ \max _ { {( x , y _ {i} ) = 0 } {i = 1 \dots n } } \ \frac{( A x , x ) }{| x | ^ {2} } ,\ \ n > 1 , \end{array} \right \} $$

where $ x , y _ {1} \dots y _ {n} $ are arbitrary non-zero vectors in $ H $. Similar relations hold for the negative eigen values $ \lambda _ {1} ^ {-} \geq \lambda _ {2} ^ {-} \geq \dots $:

$$ \tag{2 } \left . \begin{array}{c} \lambda _ {1} ^ {-} = \ \min _ {x \in H } \frac{( A x , x ) }{| x | ^ {2} } , \\ \lambda _ {n+} 1 ^ {-} = \ \max _ {y _ {1} \dots y _ {n} } \ \min _ { {( x , y _ {i} ) = 0 } {i = 1 \dots n } } \ \frac{( A x , x ) }{| x | ^ {2} } ,\ \ n > 1 . \end{array} \right \} $$

Relations (1) and (2) are applied for finding the eigen values of integral operators with a symmetric kernel. If $ A $ and $ B $ are completely-continuous self-adjoint operators, $ A \leq B $( that is, $ ( Ax , x) \leq ( B x , x ) $), $ \lambda _ {n} $ and $ \mu _ {n} $ the sequences of their positive eigen values, listed in decreasing order, where each value is repeated as often as its multiplicity, then $ \lambda _ {n} \leq \mu _ {n} $.

References

[1]	N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Wiley (1988)