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Let A be a positive Hermitian transformation on an n-dimensional unitary space En with proper values al_ >a. Let bl?-?bk be the proper values of A|M, where M is a proper subspace of En and cl.... ?c be the proper values of AIM1. Let i1 (a.-., a.)(a+,i1_1 * a,r,.-1). In this article generalizations of this inequality have been studied. Let A be a positive Hermitian linear transformation on a unitary space En with proper values a,> .. >an. Let M be a proper subspace of En. Let the proper values of AIM be bi> .. >bh and the proper values of AIM' be cl > >Ck. Then N. Aronszajn [4] has given the inequality - O for all $0. An orthonormal set {oc1,. , Ok} will be indicated by {ap,} o.n. The subspace spanned by the set {ocl, * *, oCk} will be denoted by [IX1, * *,k]. We write dim M=h if the dimension of the subspace M is h. If A is a linear transformation on a unitary space En and if M is a subspace of En, then we define a linear transformation AIM as follows: if geM, let [AIM]$=PAd, where P is the orthogonal projection on M. We observe that if oc and f3eM, then ([A|M]oc, /3)=(PAoy, f5)=(Ao, O3). It follows that if A is Hermitian (positive), then so is AIM. If jp_ip, for p=1, * * *, k, we write (jl,* jk)-< (ii, i, k) and say the sequence (il, * , ik) is greater than or equal to the sequence Received by the editors March 12, 1971. AMS 1969 subject classifications. Primary 1555, 1560.

Positive transformations restricted to subspaces and inequalities among their proper values