144 Basic Concepts of the Theory of Difference Schemes(Ay, y) = (T*STy, y) = (STy, Ty]> c 1 [[Ty][^2 ,
meaning A > / A 0 , where A 0 = T*T and / = c 1. The operator A 0 ts
self-adjoint:( A 0 y, z ) = ( T T y, z ) = ( y, T T z ) = ( y, A 0 z ).
Therefore, estimate ( 44) holds true and takes the form(65)It is worth bearing in mind here that the inverses T-^1 and ( T* )-^1 do exist.
Indeed,[[y[[~ 0 = (Aoy, Y) = [[Ty][^2 ,
Estimate (65) can be simplified in the case when the right-hand side <p of
equation (37) is of the special form <p = T* 7) and A = T* ST. The inner
product of (37) and y discovers the relationships( T S T y, y ) = ( T 7), y ) = ( T y, 7) ].
Putting these together with the inequalities(T*STy, y) > c 1 [[Ty][^2 , (Ty, 77] S [[Ty][· [[17][,
we derive the estimate 11 Ty] [ < c 1 - l [ [ r7] [.
We have restrlcted ourselves to the simplest examples demonstrating
how the a priori estimates that can be obtained through such an analysis
for the operator equation A y = <p apply equally well to important problems
arising in theory and practice.