#Java, 218 Bytes

<I>First time I've ever written Java...</I>

Golfed:

    import java.util.Arrays;boolean M(String a,String b){char[]A=a.toUpperCase().replace(" ","").toCharArray();char[]B=b.toUpperCase().replace(" ","").toCharArray();Arrays.sort(A);Arrays.sort(B);return Arrays.equals(A,B);}

Ungolfed:

    import java.util.Arrays;
    public class ManageTrashSo {
        public boolean M(String a, String b) {
        char[] A = a.toUpperCase().replace(" ", "").toCharArray();
        char[] B = b.toUpperCase().replace(" ", "").toCharArray();
        Arrays.sort(A);
        Arrays.sort(B);
        return Arrays.equals(A, B);
       }
    }

Testing:

        ManageTrashSo manageTrashSo = new ManageTrashSo();

        //True
        System.out.println(manageTrashSo.M("Lynn", "Nyl N"));
        System.out.println(manageTrashSo.M("Digital Trauma", "Tau Digital Arm"));
        
        //False
        System.out.println(manageTrashSo.M("Android", "rains odd"));
        System.out.println(manageTrashSo.M("In between days", "bayed entwine"));