On invariant subspaces of collectively compact sets of linear operators

Bir Banach uzayı üzerinde tanımlı sınırlı doğrusal operatörünün aşikar olmayan (yani ve ten farklı) kapalı (hiper)değişmez altuzaya sahip olup olmaması, ‘değişmez altuzay problemi’ olarak bilinir. Burada in bir altuzayının operatörü altında değişmez kalması, hiperdeğişmez kalması ise ile değişmeli her operatör altında değişmez kalmasıdır. Tek bir operatörü yerine bu operatörlerin bir ailesi göz önüne alındığında, nin altında (hiper)değişmez kalması, ailesine ait her operatör altında (hiper)değişmez olmasıdır. in kapalı birim yuvarını göstermek üzere, kümesi önkompakt ise ailesine birlikte kompakt denir. Bu çalışmada doğrusal operatörlerin birlikte kompakt ailelerinin (hiper)değişmez altuzayları araştırılmaktadır. Kompakt operatörlerin önkompakt ailesi birlikte kompaktır ancak bunun tersi her zaman doğru değildir. Kompakt operatörlerin önkompakt aileleri için bilinen bazı değişmez altuzay sonuçları, birlikte kompakt operatör ailelerine genişletilmektedir. Bunu yaparken, Rota-Strang spektral yarıçapı, Berger-Wang spektral yarıçapı ve deki sıfırdan farklı bir elemanı için yerel spektral yarıçapı kullanılmaktadır. Ayrıca  daki birlikte kompakt ailesinin, Berger-Wang formülünü sağladığı gösterilmektedir; burada, X in altuzaylarının tam zincirini ve, daki tüm altuzayları değişmez bırakan operatörlerin kümesini göstermektedir. Anahtar kelimeler: Değişmez altuzay, birlikte kompakt kümeler, ortak spektral yarıçap.Given a Banach space and a bounded linear operator, may or may not have a closed subspace, other than and, which is left invariant under, that is,. This work is concerned with this problem which is commonly known as the invariant subspace problem. However, instead of taking a single operator, we consider a family of linear bounded operators on infinite dimensional Banach spaces which are tied together with a strong compactness condition, known as collective compactness and we look for a common invariant subspace of elements of. A family of operators is called collectively compact if the closure of the closed unit ball of is compact under the action of. That is, is compact in. Using well-known techniques, we generalize invariant subspace results which are proven for precompact families of compact operators to collectively compact families of operators. In doing so, we use joint spectral radius, Berger-Wang spectral radius of and the local joint spectral radius for a non-zero in. A common technique to show that a multiplicative semigroup generated by has a common invariant subspace is to show that has a non-zero semigroup ideal which has a non-trivial closed invariant subspace. Employing this technique and introducing a semigroup ideal in the multiplicative semigroup, we show that if is collectively compact and then has an invariant subspace. Another results in this direction are the ones which yield a common invariant subspace for a collectively compact family of operators if for some non-zero in. If, on the other hand, is collectively compact and, then we show that has a common invariant subspace. Another case where collectively compact family has a common invariant subspace is when and is not bounded. In the final part of the work, we consider a complete chain of closed subspaces of and show that if is a collectively compact family in then we have. Here, denotes the set of operators that leave all the subspaces in invariant and denotes the set of all operators satisfying for any gap in and all. As a result of this, we relate the joint spectral radius to Ringrose?s diagonal numbers for triangularizable collectively compact sets of operators: The arbitrary complete chains of invariant subspaces for collectively compact sets of operators are then considered, and the joint spectral radius is compared with joint spectral radii of sets induced by in the quotient spaces corresponding to the gaps of the chain. So we first show that that if is a collectively family in and, then there exists only a finite number of gaps of the chain  such that. And by using this result we obtain that if is a collectively family in, then we have. We finally show that the Berger-Wang formula holds for a collectively compact family in where is a complete chain of subspaces.  Keywords: Invariant subspace, collectively compact set, joint spectral radius